Using the CLT on the logs as suggest by @kennytm might give a
satisfactory approximation. (Of course the CLT does not apply
directly to products.)
As long as we're discussing approximations, a simulation gave approximately 0.045 as the answer.
Repeated it several times to check computational stability.
One run is shown below.
(Each row of the matrix has 20 observations from UNIF(1, 2);
the penultimate statement takes products of rows; the
last statement estimates the desired probability baed on a
million performances of the experiment.)
m = 10^6; n = 20
MAT = matrix(runif(m*n, 1, 2), nrow=m)
y = apply(MAT, 1, prod)
mean(y > 10000)
Addendum: A histogram of the sums of logs of 20 obs
is nicely fit by a normal density curve. This seems to
lend credibility to the approach of using the CLT on logs.