What can I do if I'm seated next to a person with bad hygiene on an airplane?

Well,

the solution is an ointment with menthol/eucalyptus. Pathologists are using Tiger Balm, the red variant and apply it under the nose to suppress...well, I think I do not need to explain that.

It is not really necessary that it is exactly Tiger Balm, any very strong and durable menthol/eukalyptus is sufficient, you can also ask a nice pharmacist if he mixes it for you.

Some precaution: Only use as much as needed. Depending on the strength it can give a burning feeling on the skin and make your eyes and nose water.

This happens periodically, and you can politely and privately bring the problem to the attention of the cabin crew during boarding, before the aircraft door is closed, especially if the odor is so strong that many passengers are complaining. Here's an article in which one flight attendant describes one such situation. Here's another in which some possible solutions are mentioned:

• The airline can ask the passenger to clean up, even providing toiletries in some cases
• They reseat you if the flight isn't completely full
• They can spray air freshener, which might also be unpleasant for you if you are sensitive to smells
• They can kick the passenger off the flight, sometimes allowing them to fly on a later flight after they've cleaned up
• They can let you off the plane and rebook you on a future flight
• You can use a cream or balm to help reduce the odor for you, as Thorsten suggests

The response will depend on the details of the situation, the airline's policy, and the judgement of the crew on a case-by-case basis. It's important to raise the problem as early as possible, as your options become much more limited at 33,000 feet on a completely full flight.

If the aircraft is still at the gate, ask to speak to the Agent. Perhaps you can be reseated. Alternatively, the Purser might be able to assist as well.

At the Gate is the best time to address this and any similar situation as there are a lot more options available.

If the flight has departed, you easiest option is to find another open seat and move. The Flight Attendant can assist with this.

If there are no other options, adjusting the vent to blow fresh air on you or between you and the other subject may help.

I almost didn't think this deserved to be an answer, but here goes:

Have you ever attempted to ask the offending person? (I didn't mean for that to read as being confrontational, on my part.) Even if you have done so in the past, and needed to resort to other means when the accused party failed to respond, or even acknowledge your question, each person is probably not the same as those others.

Your first recourse should always be to ask the other person if they are aware that their bodily aroma could be causing offense to others.
It would work best if you didn't presume to condescend upon them as if they were a child being scolded, either: most people — those not simply indolent or inconsiderate — would take that as a sign that you are not worth a rational discussion.
Maybe they aren't aware; maybe they are aware, and have a perfectly reasonable explanation — delayed flight, inconvenient layover, temporary financial hardships, dire circumstances, refugee, so on. You might think it frustrating to listen to their arguments, yes, but this strategy applies to broader situations too: e.g. if the offending odor is not bodily funk, or even an odor at all.

The other person might be uncooperative; the manner by which they do so may be informative as to what other recourses are available to you.
Keep in mind that you can expect any perceived differences of social strata between the two of you will change your optimal approach.

Yes, in a moment of indignation, most of us tend to assume offense and to blame the other party, but unless the situation calls for such hasty responses, it is better to take a moment to fully assess all the possibilities.

Okay, I'll end so that it doesn't sound like I'm being preachy.
Anyways, this is really less of a topical answer, and more to broader interpersonal encounters anywhere, I suppose.

The gist of it is that discussing with the person might not help immediately remedy the situation between you and the other person.
Hopefully, it at least helps to ensure the second person will be easier to work with; rather than scrambling around them, confront them directly first.
It also tends to make for a better society: If the first party behaves like an adult, and treats the second party like an adult, then not only are they given more incentive to behave like one, but the first party is less likely to appear juvenile.

Find a different seat.

One way to do this is to bring $20 - $50 cash. After you board, if you happen to be in a seat you don't like (e.g. because the person next to you is smelly), find a more preferable seat that seems to have a solo passenger in it, and make an attractive cash offer to the person who currently has that seat to switch with you. Be upfront with your reason for wanting to switch, as they will likely be suspicious otherwise.

There is a simple way to reduce the impact of a strong bad smell you can do any time, anywhere.

Breathe through your mouth.

The majority of the sense of smell comes from air passing near the olfactory epithelium at the top of the sinuses, an area rich with olfactory neurons. Air breathed in through the nose passes across this area, and the olfactory neurons detect even very minute quantities of the molecules that trigger the sense of smell. The more neurons are affected, the stronger the smell.

Breathing through the mouth will greatly reduce the amount of air that flows past these neurons, reducing the impact of any smells. With conscious effort, you can effectively avoid even very strong smells this way.

The advantage of this technique is it requires no advance preparation and can be done spontaneously.

Recognize that it's your nose and tastes that are your problem, not the person sitting next to you?

You're describing a smell, but have no clue actually what is causing the smell.. Making an assumption that it's a lack of showering.. some people's sweat is pungent, but being able to smell it doesn't mean they haven't washed.

But let's go for the two most obvious..least effort solutions, shall we?

1: Travel with something you do like the smell of. 2: Ask the person to go into the toilet cubicle and have a wash, maybe you brought some soap with you? Hell, you could even pay them to do it. If that's the case then I'd agree that the person was ignorant & worthy of the kind of attitude you display. [Otherwise I'd suggest that you take a more civil approach, telling somebody they're making you want to throw up is not going to get a civil and reasoned response. Especially being as we know full well you're exaggerating for dramatic effect.]

There, that was hard.

But 3 is really the only long-term solution. Prepare yourself for life in public spaces.

You say you have an extremely sensitive nose. So the problem is you as much as it is the actual source of the smell. Well, entirely really, because you're the one that is upset by it. We can confer responsibility on others, implicitly or explicitly, but the fact is that you're the one that is suffering and you don't want to suffer. Some people take face masks into public spaces, others spray extra perfume on a stole or somesuch.

People deal with expected levels of noise pollution with earplugs, noseplugs are an obvious counter then to smells, right?

Can you tell I find this question and it's phrasing particularly odious?