I recently read about the Bailey–Borwein–Plouffe formula for calculating the $n^{\rm th}$ digit of $\pi$. I'm curious to how can we be sure that the formula is always accurate or correct?! Even if we can verify for every $\pi$ digit discovered till now with a "standard formula", how do we know that the formula is correct after those digits?
Also, what's the implication of these kind of BBP formulas for other "divine"/fundamental constants? can we learn anything else about the value of such constants and our universe through these? Any insight into the "supreme fascist" mind?
Your question makes an unfounded assumption about the nature of proof, or at least this particular proof. You say:
Your implicit assumption, it seems, is that the way one would go about verifying that this algorithm works for every single digit is to try it for every $n^{th}$ digit, and see if it really matches up to the corresponding digit of $\pi$. This would be similar to proving that $$ \frac13=0.\overline{3} $$ by checking each digit through long division and making sure that it's a $3$, or proving that one less than the square of any prime greater than $3$ is divisible by $24$ by checking each such prime. I assure you that in each case, there is a proof beyond verification. If you've never seen something similar before, I recommend that latter statement as an exercise in proof.
On top of that, you should really be asking yourself: where did we get all these digits of pi from in the first place? How do we have these $n^{th}$ digits, against which we could even check the BPP algorithm? What could anyone do to check the "standard formula"?
In any event, the reason we know that this algorithm works is that it is derived from the equality $$ \pi = \sum_{i = 0}^{\infty}\left[ \frac{1}{16^i} \left( \frac{4}{8i + 1} - \frac{2}{8i + 4} - \frac{1}{8i + 5} - \frac{1}{8i + 6} \right) \right] $$ which in turn can be derived from the equality $$ \pi = \int_0^1 \frac{4y}{y^2-2}dy-\int_0^1 \frac{4y-8}{y^2-2y+2}dy $$ Which in turn can be traced back to the nature of $\arctan(x)$, noting that $\pi = 4\arctan(1)$.
For more information on that, I recommend reading the original paper, which is surprisingly easy to follow (as far as math papers of any sort go).
As for implications regarding other numbers, there are some resulting formulae that can be reframed into similar algorithms to produce the $n^{th}$ digit of other important constants. However, this hasn't given us much information about the values of these constants as much as it's given us another way to compute them.
Still not sure where Fascism factors in, if you care to clarify...
EDIT: apparently, this is an Erdős quote. I approve whole-heartedly.