How to 'save' my U.S. mobile phone number over a year of inactivity (out of US)

Port the number to a "Virtual Number" service.

There are services such as Callhippo or Google Voice that will give you a "virtual phone number". You either access it via VoIP, or have it forward calls to another phone number.

They typically have a much more modest cost in the neighborhood of $10/month.

This has a neat feature: it means you can keep receiving calls even though you are out-of-country.

Typically you either use VoIP (e.g. through their app), or you have it forward to the physical phone that you do have in your locality. The latter works best in places where calls from the virtual number service to your phone are free.

Verizon is not going to hold your number "for free". Holding your number is a service, for which they charge money. Prepaid plans can be as little as $20/month, there's no need for a heavy data plan with all the bells and whistles.

I mean to keep the number but no service.

Aside from the virtual number services that Harper - Reinstate Monica and Midavalo mentioned in their answers, some phone plans allow pausing, such as Google Fi (which allows suspensions up to 3 months). Note that in the case of Google Fi, it can also be used outside the US.

I use both Google Fi and Google Voice:

• Upsides of Google Fi vs. Google Voice: includes mobile data in most countries, can be paused, some services (uncommon) refuse to send text messages to Google Voice numbers.
• Downsides of Google Fi vs. Google Voice: more expensive if you're already paying for another phone plan in the US (Google Voice is free unless calling/texting outside the US), can't receive call/text messages when paused, can only be used on 5 devices simultaneously (since Google Fi comes as a SIM card -> see number of data-only SIMs: "You can add up to 4 data-only SIM cards. You can also use the same data-only SIM card in multiple devices.").

You could port your number to Ting, an MVNO. It’s US$6 / month (plus some taxes) if you don’t use it.

Many of the existing answers are giving good advice about where to park your number but are glossing over the first of the two steps involved:

1. Port your number away from Verizon. As your current carrier owns your number, you need to contact them to allow the transfer away from them. Your best best is to contact their customer service department. More information may be found here: Verizon Port Out FAQ. You need to get from them an identification number proving that you are authorized to move the number, which you will provide in step 2.

2. Port your number to somewhere else. Once you have the PIN from your current provider, you can choose any number of new places to move it to, as detailed by the other answers. Costs for hosting the number will vary, as will the exact method on how you provide the PIN to your new provider.

Without doing step 1, you will not succeed in step 2.