Are you smarter than me? 13

Since $\gcd(19,92)=1$ and $\phi(92)=44$

By Euler-Fermat theorem we have $19^{44}\equiv 1 \mod 92$ $\Rightarrow 19^{88}\equiv 1\mod 92$ $\Rightarrow 19^{92}\equiv 19^{4}\equiv 361^2\equiv (-7)^2\mod 92$ $\boxed{19^{92}\equiv 49\mod 92}$

Built-in Python:

pow(19, 92, 92)

Memory-efficient method utilizing "(a x b) mod m == (a x (b mod m)) mod m":

def pow2(b, e, m):
    c = 1
    i = 0
    while i < e:
        i = i + 1
        c = (c * b) % m
    return c

print pow2(19, 92, 92)