User blog comment:Spitemaster/How big is this function based on the Harmonic series?/@comment-30754445-20190220190800/@comment-30754445-20190220210603

That's because it's not really the "slowest possible". It's just an example of an infinite sequence of functions that diverge slower and slower.

There are functions which diverge slower than all the functions in that sequence.

This isn't any different, really, from the fact that you can alway create faster-growing functions. Even if I give you an infinite family of ever faster growing functions (like x, x^2, x^3, x^4,...) you can always find a function that grows faster than all of them (like x^x).