Another way to write this sentence with the same meaning would be, "But most of all, I like to have fun".

"Most of all, however, I try to have fun" = means the same thing For example, maybe they were talking about how they are learning piano and they focus a lot on improving, but the main thing that they care about with learning it is having fun.

It’s concessive in sense (such clauses are temporal, conditional, causal, or concessive).