tmmbsl is used in instant messaging programs and emails etc. as an acronym for:

That made me big smile lol.

It is quite a complex "internet-speak" word because it incorporates an already used acronym within as well as using that nested acronym to just express humour rather than its literal meaning.

It is used when a situation is more humourous than to use tmms (that made me smile) but less humourous than lqtm (laughing quietly to myself).

tmms<tmmbsl<lqtm

Most widely used alongside an emoticon of a smile with teeth showing.

Another acronym that is part of the new wave of internet acronyms that are more truthful than current ones such as; lol, lmao and rofl.