Question: We know it converges for any prime $p$. I just want to know how to compute its exact value:
$$\prod\_{n=1}^{\infty} (1-p^{-n})$$

Answer: 
Let's fix the issue of giving bounds on the infinite product. The [pentagonal number theorem](https://en.wikipedia.org/wiki/Pentagonal_number_theorem) gives, after grouping pairs of consecutive terms with the same sign, an [alternating series](https://en.wikipedia.org/wiki/Alternating_series_test) with terms that are decreasing in modulus. So for instance for $c:=\prod \_ {n\ge1} (1-2^{-n})$ one has

$$1-\frac{1}{2}-\frac{1}{4}+\frac{1}{32}+\frac{1}{128} -\frac{1}{4096}-\frac{1}{ 32768 } < c < 1-\frac{1}{2}-\frac{1}{4}+\frac{1}{32}+\frac{1}{128}$$ that is $ 0.288787842 < c < 0.2890625, $ in any case larger than $1/4$.
The [Dedekind eta function](https://en.wikipedia.org/wiki/Dedekind_eta_function) is
$$
\eta = q^{1/24}\prod\_{k=1}^\infty(1-q^k) = 
e^{\pi i \tau/12}\prod\_{k=1}^\infty(1-e^{2\pi i k \tau})
$$
which converges for complex $q$ with $|q| < 1$. It is often written in terms of a complex argument $\tau$ with $\text{Im}\;\tau > 0$, where $q=e^{2\pi i \tau}$.
The factor $q^{1/24}$ in front gives this desirable number-theoretic properties, but clearly it can be evaluated with that factor if and only if it can be evaluated without.

Some exact values are known for the eta function (see the Wikipedia page). But not $q=1/p$, $p$ prime.