What is the maximum possible number of vertices of a $k$-dimensional polytope $P$ obtained by intersecting an $n$-vertex polytope $Q\subset\mathbb R^d$ with a $k$-flat?
For example, if $Q$ is a simplex (so $n=d+1$) then you can take a hyperplane that splits $Q$'s vertices into half and half, and you get $\Theta(d^2)$ vertices.
Consider the case where $Q$ is a simplex. Think of the $k$-flat $x_{k+1}=x_{k+2}=\dotsb=x_d=0$ and intersect it with a simplex that is intersection of the halfspaces given by inequalities $L_i(x)\leq c_i$ where $L_i$ are linear. Clearly, you can get any polytope with at most $d+1$ facets that way. The maximum is given by McMullen's Upper Bound theorem. In particular, if $k=d/2$, then it is exponential in $d$.
I do not know the answer if $Q$ is not the simplex.