I have this odd dream that online resources like this can serve as a virtual thesis advisor for future mathematicians who are teaching themselves. Here's another question along these lines. You can do a lot of different things with your day as a mathematician. Some uses of your time are better than others. The following question certainly will vary from person to person...but cogent answers to the question may teach people to do mathematics more effectively. The question is this:
What is the highest "gain" mathematical activity?
A possible interpretation of this question:
If you had to pick one thing you do as a mathematician...and it was the only thing you were going to be allowed to do mathematically for the rest of your career...and you wanted to discover/learn the most mathematics by doing it....what would it be?
Here are some candidates:
Working out a simple but nontrivial example of a theorem or question. Trying to prove a theorem for yourself in full generality. Modifying a given problem. Looking for new problems. Doing exercises. Reading books. Trying to conceptualize (making things human-friendly). Talking to people. Answering Math.Stackexchange questions. Writing things up.
As simple as this is, I'd like to see answers posted here. This is closely related to an earlier question of mine about method. (I think the way to be a mathematician is this: work really hard until stuck, learn a little bit, repeat.)
NOTE: I am not suggesting that such an activity should exist (this question is as ridiculous as the practice in philosopy of leaving all but one fact in the universe unchanged). The point is to force us to think about what such an activity would be if it should exist. I'm inclined to think that for me working on the simplest nontrivial example is best, because it forces me to connect and compare the new concept with more familiar things and simultaneously helps organize all parts of a problem into a manageable whole. I'm sure Grothendieck would not have the same answer to this question...