Highest lower bound on NP problems (TSP)
The best known lower bounds for TSP are essentially the same as those for 3SAT (within polylogarithmic factors), and they apply the fact that there are very efficient reductions from 3SAT to Hamiltonian Path. (So, a good algorithm for Hamiltonian path would imply one for 3SAT, but certain lower bounds show that sufficiently good algorithms don't exist for 3SAT.)

For the best 3SAT lower bounds known, see this answer. The bottom line is that the only non-trivial lower bounds known for these problems impose a space restriction on the candidate algorithm, as well as a time restriction. So for example, we can say TSP cannot be solved in $n^{1.5}$ time and $O(log n)$ space, but we don't know if TSP can be solved in $n^{1.5}$ time (with no space restriction).