[nmbrskeptix]

Tanh, and therefore atanh, are wonderful.

It's linear for small x, and exponential for large. Lots of applications of this:

Compressing data

Mapping (zoomed in near by, zoned out from afar)

There's a whole class of electronics amps for this.

[cyco130]

tanh is a very pleasant sounding overdrive function for audio, for example.