What are some problems in $\mathrm{P}$ with time complexity of high-degree polynomial?
Concrete example: Thorup's $O(n^{120})$ time algorithm to recognize half-squares of planar bipartite graphs.

https://arxiv.org/abs/1804.05793

Parameterized problem: Some parameterized $NP$-complete problems have arbitrarily large polynomial time algorithms. Like, finding a $k$-clique in a graph can be solved by $\Theta(n^k)$ algorithm. And we do not know how to do much better. There might be some improvement but in general when the parameter increases the degree of the polynomial run-time bound increases also. 

Computational geometry problems parameterized with the number of dimensions $d$ can be seen as a special case in this group.

Some other $NP$-complete problems (like Vertex Cover in the comment of kne below) have parameterized version with linear-time algorithm for each parameter value. So, it takes some careful investigations before naming a hard parameterized problems.