I started with (AB' + A'B)' and ended up with (A'B' + AB). Is this all the farther I can go? I feel like this is always going to be true, but I'm not sure how to prove it algebraically.
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As per request, I am making my comment into an answer.
In a sense, this is the farthest you can go. What you have reached is the equivalence operation, which is the negation of XOR. Equivalence, sometimes denoted XNOR, returns true if the inputs are either both true or both false. See the Wikipedia page for XNOR for more information.