Is it possible to blow up a balloon while you’re under water?

Yes!!! If you sufficiently increase the pressure in the balloon, it will explode(blow up) underwater.   

Or do you mean to inflate the balloon with your mouth an lungs. In which case you will very quickly have to surface for air. 

The answer is: YES

proved:

https://www.youtube.com/watch?v=wWt-B9d6l6M

(Note: go to about 8:50 to see it being done.)

Of course, how much the balloon is blown up with a single breath

depends on the size of the balloon

and on your lung capacity.

You can see in the video that the balloon used *is* blown up a little with just one breath.

You can expect to do a lot better with a small balloon.

I gotta admit that's one of the better questions i've seen in a long time.

Sure, give it a try.

Sure.   Use dynamite