Clydesdales/Athenas (200+ lb / 91+ kg) - New wheel(s) or new bike?

I bought my bike used last year. The rear wheel was badly out of true when I bought it, but I had no idea (I bought from a shop). I've had it trued and re-spoked since last June. I took it out for a ride last weekend, after 3 months off my bike, and it looks like the rear wheel needs to be trued again. I'm 5'9 and 275 lb, so I strongly suspect that the problem is the wheel itself.

I'm thinking of putting on tandem wheels, or custom wheels with 32-40 spokes, but money is tight. It's hard for me to justify spending $500 on wheels for a $350 bike. My current bike has 700cc wheels, and I want any replacement wheel to be the same size. No sense in throwing away perfectly good tires.

I've been searching the forums and the web for a bike that could support a 275 lb, 5'9" rider. I've seen some interesting prospects for $500 or less.

Alternatively, I could replace the rear wheel with a tandem wheel, and keep the same front wheel the same. That would probably cost the least.

In my situation, would you buy a new bike, 2 new wheels, or 1 new wheel?

You don't need a tandem wheel. You need a wheel builder that knows what he or she is doing. I would suspect any bike shop that would let a bike leave their shop with an out of true wheel, especially to a large rider. They had to have known that bike was coming back. Since you are short on finances I would suggest you look for a different bike shop. Look for one with an experienced wheel builder. Have them go through your wheels. They should be able to advise you on what your next step should be but it should be along the lines of going though and properly tensioning your current rear wheel (unless there are fatigue issue with the rim or hub).

None of the above. Build a wheel yourself. Over in the mechanics or C&V forum, we'll teach you how to build your own wheel. It will last a good, long time.