The greatest common divisor of $ 2^a3^b5^c\cdots $ and $ 2^{a'}3^{b'}5^{c'}\cdots $ is $$2^{\min\{a,a'\}}3^{\min\{b,b'\}}5^{\min\{c,c'\}}\cdots.$$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{align*} \gcd(&2^23^35^57^711^{11}13^{13}17^{17}19^{19}23^{23},\\ &\quad 2^{23}3^{19}5^{17}7^{13}11^{11}13^717^519^323^2) \\=\ & 2^23^35^57^711^{11}13^717^519^323^2.\end{align*} So, the prime with largest exponent in the $ \gcd $ is $ \boxed{11} $.