Decision problem that can be verified but not run in n^2 time
The answer depends a lot on the model of computation.

On one extreme: in the decision tree model, many lower bounds can be proven.  For instance, consider the 3SUM problem.  You can verify an alleged solution to the 3SUM problem in $O(n)$ time, but it's conjectured that no algorithm can solve the 3SUM problem in $O(n^{2-\epsilon})$ time for any $\epsilon&gt;0$.  One can prove this holds in some version of the algebraic decision tree model; one can prove that any algorithm to solve it in this model must take $\Omega(n^2)$ time.  This provides a gap of $O(n^2)$ vs $O(n)$ for solving vs verifying solutions, if the verifier is limited to the algebraic decision tree model.  This is a pretty limited result; in this case, the model means that we're restricting attention to algorithms of a particular form.  So, it doesn't rule out the possibility of some other algorithm (that does something weird) being faster.

On the other extreme: if we allow arbitrary (non-uniform) boolean circuits, then there is no explicitly stated function $f$ on $n$ bits where we can currently prove that every circuit for computing $f$ needs $\ge 3.1n$ gates.  In other words, this is saying that we have no clue how to prove lower bounds for circuits.  Roughly speaking, we have no known result of a problem where we can prove that solving it requires circuits of size $\omega(n)$.  See, e.g., https://cstheory.stackexchange.com/a/8005/5038 and https://cstheory.stackexchange.com/q/21400/5038.  So, given our current state of knowledge, we have no hope of proving a result like that if the model of computation is unrestricted boolean circuits.