jmc

The greatest common divisor of $2^a{3}^b{5}^c\cdots$ and $2^{a^{\prime }}{3}^{b^{\prime }}{5}^{c^{\prime }}\cdots$ is $$2^{\min \{a,a^{\prime }\}}{3}^{\min \{b,b^{\prime }\}}{5}^{\min \{c,c^{\prime }\}}\cdots .$$That is, each prime occurs with the smaller of the two exponents with which it occurs in the prime factorizations of the original two numbers.

In this case, the prime $11$ occurs with exponent $11$ in both of the original numbers, while all other primes occur with an exponent smaller than $11$ in one of the original numbers. Specifically, $$\begin{array}{rl}\gcd (& 2^2{3}^3{5}^5{7}^{7}11^{11}13^{13}17^{17}19^{19}23^{23},\\ & 2^{23}{3}^{19}{5}^{17}{7}^{13}11^{11}13^{7}17^{5}19^{3}23^2)\\ =& 2^2{3}^3{5}^5{7}^{7}11^{11}13^{7}17^{5}19^{3}23^2.\end{array}$$So, the prime with largest exponent in the $\gcd$ is $\boxed{11}$.