Is there an "internet anywhere" device I can bring with me to visit the developing world?

I suspect you may be barking up the wrong tree here. In short, satellite Internet is likely vastly more expensive and limited than you expect, while regular old cellular networks are likely far more accessible that you'd expect in the developing world. This may change in a few years when high-speed satellite networks like Starlink start rolling out, but as of 2020 we're not there yet, and even Starlink requires a bulky fixed dish antenna.

For satellite internet, you could purchase something like the Iridium Go satellite wifi hotspot. However, you're looking at $700 for the device and $150/month for a data plan, and while this doesn't sound too bad, your Internet access is severely limited to an intentionally crippled custom app that, to quote a reviewer, "made me want to beat this thing to death with a chair." In addition, satellite phones are heavily restricted in many countries, notably India where there's a blanket ban unless you apply for special permits.

By comparison, for plain old mobile data:

• India: Jio has 4G coverage across virtually all populated areas of India. Network congestion can be a problem and electrical supplies in rural areas (and some cities) are also flaky.

• "parts of Asia": In all populated parts of Asia (except perhaps North Korea), including China, Japan, Korea and SE Asia, fast & cheap mobile coverage is ubiquitous; if anything, service is likely better than what you're used to at home. You'll get the absolute cheapest prices by getting a new SIM in each country, or you can pay a bit more to minimize hassle with something like SingTel's ReadyRoam that covers 9/18/81 countries at various price points.

• Sub-Saharan Africa: This is going to be the biggest challenge, with wildly varying coverage and pricing. That said, even here any city of any size will have coverage, although it may be expensive and censored, particularly in countries with tightly controlled state monopolies (eg. Ethiopia).

In addition to mobile data, Wifi is table stakes at any self-respecting hotel/hostel/guesthouse/Airbnb pretty much anywhere in the world, so if you choose accommodations that offer this (and in Asia it's really quite hard to find a place without it), you'll have a connection that's faster, more reliable and cheaper yet than mobile data.

You're overlooking the fact that mobile telephony is a far more efficient way of connecting people in remote areas than is wiring up everyone's home. Governments and telecommunications companies, at least, have not overlooked this, so cellular connectivity is generally good in towns everywhere. You should certainly expect to find good cellular service and internet in any place that is politically stable, and you probably won't want to spend too much time in the places that aren't.

In Africa, I've been to Dakar, Senegal and Kigali, Rwanda (in 2016) and to Johannesburg and Cape Town (in 2003), and in none of those places will you have trouble finding internet access. I did the most traveling in Rwanda. WiFi was ubiquitous even in smaller Rwandan towns, in hotels and cafes. Cellular service was generally excellent. Down the street from my friend's house in Kigali was a co-working space that offered internet connectivity and refreshments.

I wouldn't be surprised if the situation is different in other parts of Africa, but a quick look at the communications section of each country's CIA World Factbook page will give you some idea of how different. For example, Rwanda has 76 mobile telephone subscriptions per hundred inhabitants, while neighboring Democratic Republic of the Congo has 43 and nearby Kenya has 104. In contrast, the number of landline subscriptions in these countries is under 1 per hundred people.

For a qualitative comparison, have a look at the "general assessment" subsection. Rwanda:

govt. invests in smart city infrastructure; expanding wholesale LTE services; govt. launches SIM card registration; growing economy and foreign aid help launch telecom sector, despite widespread poverty; slow to liberalize mobile sector; competing operators roll out national fiber optic backbone that connects to submarine cables of neighboring countries ending expensive dependence on satellite (2020)

DRC:

poorly developed national and international infrastructure; bandwidth is limited; Internet pricing is expensive; domestic satellite system with 14 earth stations; wars and social upheaval have not promoted advancement; a revised Telecommunications Act adopted in May 2018; govt. only loosely regulates the telecom sector, much of the investment is from donor countries (specifically China) (2020)

Kenya:

the mobile-cellular system is generally good with a mobile subscriber base of 47 million, especially in urban areas; fixed-line telephone system is small and inefficient; trunks are primarily microwave radio relay; to encourage advancement of the LTE services the govt. has fostered an open-access approach and pushed for a national broadband strategy; more licensing being awarded has led to competition which is good for growth; govt. commits KE 300 million to its free Wi-Fi project (2020)

Unless you have very adventurous travel goals, you're very unlikely to find yourself in a place where satellite makes more sense than cellular or public WiFi.

The other answers highlight why mobile networks might actually be your best bet for connectivity. LTE is available in all but the most remote places. I'd only consider satellite service as an emergency backup for when you are too far remote to get mobile coverage.

The question then becomes how to get access to mobile networks as you travel.

The traditional answer to this is to buy local SIM cards in the countries you visit. This definitely works, can be very cheap, but can be painful to manage. Some countries have instituted strict identity requirements to activate SIM cards that can be difficult for foreigners to navigate.

An alternative that has become more workable in recent years is a phone plan that includes global roaming. These plans work in a long list of (but not all) countries, and generally include data at a flat rate no matter where you are. A single SIM means you don't have hunt down cards when you arrive in a new country, and you'll have a single phone number that you'll always be able to receive calls & texts on (which might be important if your boss needs to call you).

My pick for this is Google Fi. One rate gets you coverage almost everywhere, and you get free data-only SIMs, so you can put one in your phone and one in your LTE-equipped laptop or hotspot.

This could be supplemented with local SIMs where necessary.

As a software engineer who needs to stay productive as a nomad, it's probably more important that you plan ahead and prepare for spotty connectivity and unfortunate events.

• Have a big hard drive and store as much as you can locally. Documentation for all the libraries you use. Local cache of your package/dependency manager(s). (eg) Avoid the cloud as much as possible: you should be able to work effectively even when fully offline. Even when you are online, you don't want to have to sit and wait on large downloads.
• Run on the assumption that any or all of your equipment will be lost, damaged, or stolen. Full device encryption is critical. (Stolen API keys would be very not good!) Multiple backups! In the cloud and on flash drives. Have a plan for device replacement — it can be hard to source quality hardware quickly in some places.
• Have a plan to get in contact with work, friends, and family even if you've lost everything in your possession. ie some low-value, limited account you can log in to from an internet cafe.

To expand on the "cellular" part of the equation, I use a device from Skyroam (no affiliation) which basically gets a virtual local sim for you. It's not cheap if you want a lot of data, but it's great for checking your email and generally staying connected. Unlike a sim solution, you don't have to find out where to get a sim card, put the sim card into your phone, type long strings of numbers etc, manage old cards that might use again, and so on. You just power it on and connect to it as a wifi hotspot. Also you pay the Skyroam people so you don't need to investigate which local provider offers the better deal. If a place has cell service, the Skyroam works. It even works in Singapore, where all cell phones are connected to identity and tourists have to show their passports (and the number is recorded) to activate a temporary sim. I've used it on trains and in the back of cabs and at campsites -- all places where wifi is pretty unlikely.

Note: it does not get you a phone number. Services like texting that rely on a phone number won't work. It gets you a wifi hotspot.

I am not going to link to it because I don't want to spam.

Country Coverage (copy-pasted from website and formatted in Notepad++):

The Americas: Anguilla, Antigua & Barbuda, Argentina, Aruba, Bahamas, Barbados, Belize, Bermuda, Bolivia, Bonaire, Brazil, British Virgin Islands, Canada, Cayman Islands, Chile, Colombia, Costa Rica, Curacao, Dominica, Dominican Republic, Ecuador, El Salvador, Grenada, Guadeloupe, Guatemala, Guyana, Haiti, Honduras, Jamaica, Martinique, Mexico, Montserrat, Nicaragua, Panama, Peru, Puerto Rico, Saipan, St. Kitts & Nevis, St. Lucia, St. Marteen, St. Vincent, St. Vincent & the Grenadines, Suriname, Trinidad & Tobago, Turks & Caicos Island, United States, Uruguay, Venezuela

Europe: Albania, Austria, Belarus, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Liechtenstein, Lithuania, Luxembourg, Malta, Monaco, Montenegro, Netherlands, Norway, Poland, Portugal, Romania, Russia, San Marino, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Ukraine, United Kingdom, Vatican

Asia: Bangladesh, Brunei, Cambodia, China, Hong Kong, India, Indonesia, Japan, Kazakhstan, Kyrgyzstan, Laos, Macao, Malaysia, Myanmar, Nepal, Philippines, Singapore, South Korea, Sri Lanka, Taiwan, Tajikistan, Thailand, Uzbekistan, Vietnam

South Pacific: Australia, Christmas Island, Fiji, Guam, New Zealand

Africa: Egypt, Kenya, Madagascar, Mauritius, Morocco, Nigeria, South Africa, Tanzania, Tunisia, Zambia

Middle East: Bahrain, Dubai, Iraq, Israel, Jordan, Kuwait, Oman, Qatar, Saudi Arabia, Turkey

Something I have done in Thailand out in the country is I took an unlocked MiFi unit (purchased cheap on eBay) with me and get a local operator SIM card to put in it when I get there, then I taped the MiFi to the end of long bamboo and secured the bamboo so the MiFi was 4 or 5 metres or more in the air for a better signal, and I connected to WiFi it provided. If I wanted the MiFi to run for longer periods than its battery could support I ran a long USB cable from a USB charger up the bamboo to keep it charged up. Check the mobile network frequences and such to ensure the MiFi is compatible with the network.

My answer is not based on the developing world, but I'll post it anyway since I did this myself for 2 months in Europe and some of the tricks I discovered may still be useful to you. You didn't specifiy whether you will be travelling with a vehicle: in my case I had a car and a very large tent which I used as my "office". In no particular order, these are the things I did. Some of these may seem extreme but it was vitally important to me to maintain connectivity during working hours due to the need to be in multiple Zoom meetings every day, so I erred on the side of caution. In practice I rarely needed to use everything together as I mostly found the 4G to be reliable in the areas I visited.

Internet

• I purchased a good quality unlocked 4G router with good reviews and with the option to switch between built-in antennas and an external atenna (in my case, a Huawei B535-232). I researched and ensured that it would cover the frequency bands in use in the countries I was visiting.
• I purchased a good quality external omni-directional 4G antenna (Poynting 4G-XPOL-A0001). If you wanted to cover all bases you could also purchase a directional version. These need to be fine tuned so that they are pointing towards the base station. They can theoretically perform better than omnis, whereas omnis are more reliable and general purpose.
• I purchased a 1.5 - 4m telescopic pole to which the antenna could be attached, an auger and a portable battery operated drill/driver to make holes in the ground for the pole (in my case this doubled up as a drill for special-purpose tent pegs which saved me a lot of time in setup and teardown), and a fillable parasol base for stablisation which I filled with sand and water, attached a sponge to the hole with a glue gun, and then drilled a new hole to fit the pole snugly. The top parts of the pole were secured to a tent/tree/building etc. with cable ties. If you're operating out of buildings, you can attach the antenna to a roof or outside a window (the Poynting antenna came with various attachments for doing this).
• I brought various SIM cards from my home country with good roaming packages, and also purchased a local SIM with a good data package in the destination country. I researched the routes I would be taking in advance, checking 4G coverage maps, and trying to ensure that I had a SIM card that would connect to each of the local networks so I could choose freely between them at each spot in case of differences.
• At each location, before deciding to stay in a place I used https://www.speedtest.net/ to check that the 4G speeds were sufficient. I considered anything consistently around 10Mbps up + down to be good enough to reliably handle Zoom-style meetings (see bandwidth requirements). You can get by on less but I wanted to be sure. If I got something slow or inconsistent, I moved on.
• Once established at a site and if setting up the antenna (which was rarely necessary), I logged in to the router and checked the signals strengths against a chart such as the one below. I then made my adjustments (internal vs external antenna, height, direction, etc.) to find the most optimal setup:

Electricity

• I only stayed in cabins or campsites with electricity, and brought a camping mains lead and an adapter, having researched first what connections would be needed. I also brought a polarity tester which helped me quite a few times when I had issues.
• I brought multiple high capacity and good quality USB portable chargers (to ensure a reliable source of 4G tethering from my phone) and laptop portable chargers. Anker and Ravpower are my preferred brands - any good one will do but avoid the cheap unheard of brands. The laptop charger I have linked to provides my laptop with around 3 hours of extra charge for light usage, but be very careful to check that it is rated to handle the power requirements of your laptop. I made a mistake with lending one to someone with a higher powered laptop than mine and it stopped working for several weeks. More expensive higher capacity portable laptop chargers are available on the market, but that one was adequate for my needs as I was rarely away from electricity for long enough for it to matter.
• I brought plenty of multi-socket mains leads and good quality quick charge multi-port USB chargers so that when I had access to electricty I could charge everything at once as quickly as possible. I also brought a quick charge multi-port USB car charger so that I could rely on my car when all else failed. I even brought a portable car battery jump starter in case I accidentally drained the car battery. This also doubled up as an additional last-resort USB charger.
• I brought spares of all charging and other cables to allow for things to go wrong.
• I also bought a solar powered USB charger but never ended up needing it.

4G tests

Before I set off, I tested my setup in various parts of my house: (1) inside, (2) outside with the building blocking the path to the base station, (3) outside on the side of the base station, and (4) on the top floor near a window. At each spot I tested: (a) 4G tethering to my phone, (b) 4G via the router with built-in antenna, (c) 4G via the router with external antenna.

What I discovered was that no setup is perfect for all scenarios. In some locations, the router performed better with the built-in antenna. In some the external antenna was noticeably faster. Generally, the 4G router outperformed my phone (which is a high end model) regardless. By tinkering with the setup (including adjusting the height and direction of the external antenna) I was almost always able to get some improvement. In the best cases the optimal setup was around 10x faster than the least optimal one.

Solution I did not use

I spent a lot of time researching 4G signal repeaters (e.g. https://mobilesignalrepeater.co.uk). These essentially combine the ideas above (4G router + antenna) with a device that broadcasts a new local 4G signal. In other words, it acts like a mini base station. In the end I decided not to get one due to legal issues (various models may be illegal depending on the country) and cost (the ones that cover all 4G bands are very expensive).

Closing remarks

I thoroughly enjoyed travelling and working. In my case I worked during the daytime Monday-Friday and spent my evenings and weekends in moving around and exploring/visiting. It's incredibly lucky to have work which allows for this possibility and in my opinion you should jump on the chance as you may not be so lucky in future. I have no regrets and definitely plan to do it again.

Update on 2022-02-02 regarding starlink, from https://www.cnbc.com/amp/2022/02/02/spacex-starlink-premium-satellite-internet-tier-at-500-per-month.html:

SpaceX says Starlink Premium is capable of connecting from anywhere. Starlink Premium also offers "unlimited service locations" flexibility. The first premium deliveries will begin in the second quarter [of 2022].

Original answer:

The AST SpaceMobile Investor Presentation released on 2020-12-15 (mirror) gives a nice synopsis of the satellite-based “internet anywhere” solutions:

The AST SpaceMobile network is scheduled to be progressively deployed over 2022-2024, with the first target being the 49 largest countries in the equatorial regions (mirror).

FYI: Portable Starlink antennas.