I didn't ask for this! A person would say this if something more than they expected happened to them.

To "ask" means to request an answer. To "ask for" is to request something, that is, whatever comes after "for". So you could "ask for an answer" and that would be the same as if you "ask". "I didn't ask for this," literally means "I did not request this." You may or may not have wanted "this", it just depends on the situation. It also does not mean "I did not ask this question." "I didn't ask this," is probably not something you would say. "I didn't ask that," would be the commonly used phrase, meaning "I did not ask that question." (You could say "I didn't ask this" if you were directly indicating the question that you did not ask. For example, if you had written out a question on a piece of paper and you were pointing at the paper, you could then say "I didn't ask this.")

I didn't ask FOR this = I didn't want this. I didn't ask this = I didn't ask this question