Is there a maximum amount of antibodies your body can keep?
Antibodies are simply proteins and like any other protein have a relatively short "life", so after clearing out an infection, they are not retained for long (most of them anyway). What the body keeps is memory cells which can produce a much more rapid response if they come in contact with the same pathogen again.

You could see it as a selective process: the body produces immune cells, one or more specific cells for nearly every potential antigen that might exist in the world (there are more steps involved of course but they are of little relevance here). Only those which are used at some point mature into memory cells.

Unfortunately, I just realised that I can't explain why the immune system goes through this selective process; producing cells equivalent to memory cells in the first place would make the immune response much stronger. One explanation I can think of for why the body produces naive cells first is that either a) their production or b) their maintenance is less costly.

The other explanation I could think of is that even immune cells in the periphery which have undergone the proofing mechanisms of immune cell development already, may not be perfect and target body cells every now and then - in that case it would be devastating if they would go all-out like a memory cell.

From my three explanations, only alternative b) would mean there could be a limitation to how many vaccinations our immune system can bear (if memory cells take more effort to maintain than naive cells, having too many of them may overstrain whatever systems maintain them). Otherwise I don't see any limiting factor for how many memory cells you can retain; except if their numbers become so large that your lymph nodes swell and that causes problems.

Edit note: I'm not aware of any research about that question and couldn't find anything either.