> So my question is, how hard of a problem is that with LLMs? I get the sense that LLMs are trained on a very large data set all at once, but that it is difficult to incrementally update them with new data.

It's the opposite. Getting an LLM to learn the basics of grammar and paragraph-level language structure is the "hard" part. Once you have that, further fine-tuning, specialisation, or other incremental changes are comparatively easy.

Catching up to current events could be done in almost real time, it's "just" continuous training.

The only challenge is trying to do that and also have an LLM that's filtered to exclude profanity, racism, etc...

That filter is usually added on as a final supervised training step, and requires many man-hours to train the AI to be well behaved.

I suspect that it would be possible to automate the filtering by making another AI that can evaluate responses and score them bases on profanity level.