Are there any examples of this? If its half-life is 6 hours, then after six hours half of the caffeine is still there, and we know caffeine has a half life of 6 hours. I don't know of any molecules with a non-linear half-life but I might be completely wrong

There's a whole area for this - pharmacokinetics. See e.g. https://en.wikipedia.org/wiki/Biological_half-life#Rate_equa... for examples of e.g. first order models (exponential elimination rate, etc.) It has to do with protein binding dynamics among other factors I have no idea about. Remember that you are dealing with biological systems.

Some of these processes to do with elimination and clearance are very much non-linear and are modelled with diff equations. https://en.wikipedia.org/wiki/Physiologically_based_pharmaco...

There aren't many with zero-order pharmacokinetics (linear curve decay). The only notable exception is alcohol though.

Interesting, for a second I thought that had to be wrong because I was aware of aspirin and alcohol, but apparently they are the two common ones and the vast majority are a bit more complex, which also makes sense if you think about it logically I suppose.

It looks like in addition to those, others with zero-order pharmacokinetics include salicylates (salts of salicylic acid incl. aspirin), omeprazole (for indigestion), fluoxetine (antidepressant), phenytoin (for epilepsy), methanol (a poisonous alcohol) and cisplatin (chemo medication.)