Shortest formula for an n-term monotone CNF
You can get an $\Omega(n^2/\log n)$ lower bound using a counting argument: there are $exp(n^2)$ $n$-term CNFs on $n$ variables (this is easy but requires just a little care as to make sure that there is no over-counting), but there are $exp(s \log s)$ formulae (or even circuits) of size at most $s$.  

It seems that winning the last $\log n$ factor will be hard since it would require proving a quadratic lower bound on formula size, and my guess is that the few existing techniques for this don't suffice.  $O(n^2/\log n)$ may even be the right answer -- at least for circuit size -- using some four-Russian-like technique.