Finite One-Way Permutation with Infinite Domain
The paper On constructing 1-1 One-Way Functions, by Goldreich, Levin and Nisan shows how to construct length preserving 1-1 functions with infinite domains and finite description. The hardness of inverting the functions is based on popular assumptions, such as hardness of inverting RSA or finding Discrete Logarithms. 

Their construction is a twist of the straightforward idea of converting a family, $\{f_i\}_i$, of one-way functions into a single one-way function by setting $f(r,s) = f_i(x)$ where $r$ is the randomness used to pick the index $i$ and $s$ is the randomness used to select the input $x$ (given the index $i$). 

The problem with the above idea is that $f(r,s)$ is not necessarily 1-1. They amend this problem by slightly modifying $f(r,s)$ and arguing that, under certain conditions on the family $\{f_i\}_i$, the new construction is indeed 1-1. They then go on to show that these conditions are satisfies by the RSA/Discrete-log based functions.