This is why you care about the lower bound:
If I say something is O(n^n)), you may come back and say "I wonder if it is O(n^2)
If you say Omega(n^2), I know not to bother.
If I say something is O(n^n)), you may come back and say "I wonder if it is O(n^2)
If you say Omega(n^2), I know not to bother.
I added the observation that even theorists - the people who are experts in this area - talk like that, and gave a possible reason why. My reason, which perhaps I did not communicate well, is that the lower bound is generally not of practical interest.
For some problems, the best known upper bounds don't match the best known lower bounds. Also, in cases where there are further improvements in the upper bounds, it doesn't automatically mean people stop talking about previous upper bounds.
(Ignore this whole comment if you specifically meant the function n^n, I have no examples relevant to that function.)
I have discussed matrix multiplication's unknown lower bound with theorists, but more as a trivia item because there is a relatively recent result showing that it is O(n^2.3727) (http://www.cs.berkeley.edu/~virgi/matrixmult.pdf). That is, the unknown lower bound was the point of the conversation, and it was interesting because the lower and upper bound are not the same; it's still open research. The other point the theorist made during that conversation is similar to what I've said: getting a tighter bound on matrix multiplication is theoretically interesting, but there's not much practical application.
I was rather talking about when we discuss algorithm when trying to solve problems.