Maximum value of negligible probability?
It depends on context.

In a theoretical context, baring other definition, a probability is negligible if it decreases exponentially (or: faster than any polynomial of fixed degree) with some security parameter $k$, often a number of bits in the key; while the computational effort to use the system grows no faster than some polynomial of $k$.

In a practical context, if the probability is for one try of the adversary, and more tries will increase that probability about linearly, something like $2^{-80}$ used to be considered adequate, $2^{-128}$ is considered adequate for a decade or two baring hypothetical future quantum computers usable for cryptanalysis, $2^{-192}$ and $2^{-256}$ are sometime specified (because of security classification levels, and because that improves confidence when considering said quantum computers). Lower values will do, of course. Higher values (like $2^{-64}$) wight still make sense if the stakes are low and lowering that probability was costly, or forbidden by law.

In a practical context, if the probability is with all the parameters influencing it (effort of the adversary, number of users, size of plaintext..) set to credible values maximizing the probability, then $2^{-10}$ to $2^{-40}$ is reasonable, depending on how damaging the low-probability event would be. Lower values will do, of course.