How small is negligible? When proving theorems in crypto we often make use of the concept of negligible functions or, more simply, negligible parameters.

As a rule of thumb, given today (2018) computational power, what is the smallest inverse power of 2 that we can consider negligible?
I expect this to be something around $2^{-64}$.
Negligible is a human term, not a precise definition.  It refers to things which are sufficiently small that one is willing to ignore it in the interests of expediency.  The threshold varies from person to person: a NSA mathematician will have a different threshold for a property than a CEO of a startup will).  It also varies from topic to topic: $2^{-64}$ solar masses is still a solid quarter of the entire biomass of humans put together!

In my experience, the proof of negligability is often not how small the number is, but how large its inverse is.  Its inverse is often highly associated with some real practical number, like how much energy it takes to crack an algorithm, or how many years.

For example, the risk associated with the brute forcing of a 256-bit key is negligibly small.  This is because we can calculate how much energy it would take to count that high, much less run the algorithm that many times.  I did the numbers once, with ultra-idealized hardware running in the cold of space, it takes about 3/4 of the energy in the galaxy to run that counter from 0 to $2^{255}$