Please explain what's wrong with the proof that every group element is its own inverse.

What is wrong with my proof here?

Proof:

Let $a, b$ be elements of a group and let $aa = b$.

Through manipulation, we see that
$$a = ba^{-1}$$
$$b^{-1}a = a^{-1}$$

$$b^{-1}aa^{-1} = e$$
$$b^{-1} = e$$

We of course know that
$bb^{-1} = e$
And from above, we see that
$be = e$
or
$b = e$.
And since $aa = b$, we can see that $aa = e$ and that $a=a^{-1}$.