jmc

Notice that every term $b_n$ will be a power of 2, the exponent of which will be the sum of the exponents of the two previous terms. Therefore, let us construct a sequence $a_1,a_2,\dots$, such that $a_1=0$, and $a_2=1$, and $a_{n+1}=a_n+a_{n-1}$. Of course, $a_{20}$ is simply equivalent to the 19th term of the Fibonacci Sequence, 4181. Thus, $b_{20}=2^{a_{20}}=\boxed{2^{4181}}$.