Super-polynomial time approximation algorithms for MAX 3SAT
One can get a $7/8+\varepsilon/8$ approximation for MAX3SAT that runs in $2^{O(\varepsilon n)}$ time without too much trouble. Here is the idea. Divide the set of variables into $O(1/\varepsilon)$ groups of $\varepsilon n$ variables each. For each group, try all $2^{\varepsilon n}$ ways to assign the variables in the group. For each reduced formula, run the Karloff and Zwick $7/8$-approximation. Output the assignment satisfying a maximum number of clauses, out of all these trials.

The point is that there is some variable block such that the optimal assignment (restricted to that block) already satisfies a $\varepsilon$-fraction of the maximum number of satisfied clauses. You'll get those extra clauses exactly correct, and you'll get $7/8$ of the the remaining fraction of the optimum using Karloff and Zwick.

It is an interesting question if one can get $2^{O(\varepsilon^2 n)}$ time for the same type of approximation. There is a "Linear PCP Conjecture" that 3SAT can be reduced in polynomial time to MAX3SAT, such that:


if the 3SAT instance is satisfiable then the MAX3SAT instance is completely satisfiable,
if the 3SAT instance is unsatisfiable then the MAX3SAT instance isn't $7/8+\varepsilon$ satisfiable, and
the reduction increases the formula size by only a $poly(1/\varepsilon)$ factor. 


Assuming this Linear PCP Conjecture, a $2^{O(\varepsilon^c m)}$-time $7/8+\varepsilon$ approximation, for all $c$ and $\varepsilon$, would entail that 3SAT is in $2^{\varepsilon n}$ time, for all $\varepsilon$. (Here $m$ is the number of clauses.) The proof uses the Sparsification Lemma of Impagliazzo, Paturi, and Zane.