It is not possible to have a simple graph with all distinct degrees.

It can be proven with the pigeon hole principle.

For a graph with n vertices:

Case 1: One vertex has degree 0. This means the degree for vertices range from 0 to n-2, and so 2 of n vertices must have the same degree by pigeon hold principle.

Case 2: All vertices has degree > 1. This means that degree for vertices range from 1 to n-1, and again 2 of n vertices must have the same degree by pigeon hole principle.