tldr: the number of species grows logarithmically with space and time.

Can't help but think of a species as a tree in a decision tree forest. Each tree is concerned with a niche of the ecosystem and optimizes to reduce entropy in that segment. Given more time (training data) we can have more specific branches. Given more space we can have more decision trees.

Isn't pretty much all growth inherently logarithmic with time?

Wait, why? Isn't most growth exponential with time (until it crashes)?

It's only exponential until countering factors come into play, and they certainly don't have to cause a "crash".

Power law, exponential... Was using the parent comment's term but we mean the same thing here.