Orbital decay via gravitational emission in a hypothetical, two-body universe
Using this equation from Wikipedia for the approximate merger time:
$$\frac{5}{256}\frac{c^5}{G^3}\frac{r^4}{m_1m_2(m_1+m_2)},$$
and Wikipedia's values for the Moon's equatorial radius and mass, and the launch mass of the HST, we can use this query in Google:
((5/256) * (c^5/G^3) * ((100 miles) + (1738.1 km))^4 / ((11110 kg) * (7.347631E22 kg)^2))
to perform the calculation.
The result is $\approx 3.4497 \times 10^{46}$ seconds, or around $1.093 \times 10^{39}$ years, which is about $7.93 \times 10^{28}$ times the current age of the universe.
That equation is for a circular orbit, and it's only a crude estimate based on orbital energy lost via gravitational radiation. It mostly ignores the change in the orbital radius, since that's fairly small until the final stages of the merger.
There's some information about the effect of orbit eccentricity  on changes to orbital angular momentum and energy due to gravitational radiation at Two-body problem in general relativity.