Power tower modulo $n$

Reduce exponent. The exponent is $2^X$ where $X=2^{2^{\cdots}}$ (with $2023$ twos). Since $\gcd(2,10)=2$, this is tricky. Instead, compute: $2^{2^k}\pmod{10}$ for $k\ge1$: $2^2=4$, $2^4=16\equiv6$, $2^6=64\equiv4$ — period $2$. For $k\ge2$ even, $2^{2^k}\equiv6\pmod{10}$. So for a tower of $2023$ twos (exponent), $2^{\text{exponent}}\equiv2^6=64\equiv9\pmod{11}$.

Simpler. Since $2^6=64\equiv9$ and $2^{10}\equiv1$, the answer is $9$ for $2024$ twos (because $2^{2^{2}}$ with $3$ twos is $2^4=16\equiv5$, $2^{2^{2^2}}$ with $4$ twos is $2^{16}\equiv2^6\equiv9$, and it stabilizes at $9$).

Conclusion. $2^{2^{\cdots}}\equiv9\pmod{11}$ for a tower of $2024$ twos.

Caveat: This requires careful analysis of the exponent's residue mod $10$; details depend on exact tower height.