Is there a way to make a unique association between a natural number with leading zeros with a natural number that has no leading zeros?
I have thought about this question for long enough to conclude that it seems impossible, although I am not 100% sure.
N.B. I am talking about decimal naturals.
I suppose that by unique association you mean a bijection. Setting $A = \{0, 1, \ldots, 9\}$, $B = \{1, \ldots, 9\}$ and adopting the notation used for regular languages, the set of natural numbers with leading zeroes is $L = 0^+A^*$ and the set of natural numbers without leading zeroes is $N = BA^*$. Since both sets are countable, there is certainly a bijection between them.
One way (probably not the simplest one) would be to order lexicographically the elements of $L$ and $M$. This would give two bijections $f:L \to {\Bbb N}$ and $g:M \to {\Bbb N}$ associating to each element its lexicographic rank and it just remains to consider the bijection $g^{-1}\circ f: L \to M$.