Paper about the upper bound of the number of inequalities to describe the Integer hull of a polyhedron
Via Rubin:

There is a sequence of polytopes $\{ P_k\}_k$ each defined by two variables and one constraint (plus two nonnegativity constraints) such that the integer hull of $P_k$ has $k+3$ vertices and $k+3$ facets. 

$P_k=\{ (x,y)\in \mathbb{R}^2_+ ~|~f_{2k} x + f_{2k+1}y \le f^2_{2k+1}-1\}$

Here, $f_j$ is the $j$-th Fibonacci number.

This shows that the number of facets of the integer hull $P_I$ cannot be bounded above in terms of the number variables and constraints of $P$.

Schrijver provides a nice collection of these kind of results in his Theory of Linear and Integer Programming.