Could it be possible to train LLM only through the chat messages without any other data or input?

If Guppy doesn't know regular expressions yet, could I teach it to it just by conversation? It's a fish so it wouldn't probably understand much about my blabbing, but would be interesting to give it a try.

Or is there some hard architectural limit in the current LLM's, that the training needs to be done offline and with fairly large training set.

What does "done offline" mean? Otherwise you are limited by context window.