Raising both sides to power of $88$ and re-arranging the left side gives: $$\left(x^{88}\right)^{\left(x^{88}\right)}=88^{88}$$

Let $t=x^{88}$. Then $t^t=88^{88}$ and $t>0$. We argue that $t$ must equal to $88$. If $t > 88$, then $t^t > 88^{88}$. Meanwhile, if $t < 88$, then $t^t < 88^{88}$.

Therefore, the solution to the original equation is $x=\boxed{\sqrt[88]{88}}$.