I assume by LLMs you actually mean "ChatGPT and similar commercial LLM services". Raw LLMs like LLaMA and Falcon will happily spout out the craziest, sometimes super creative ideas, it's just the lobotomising of ChatGPT and the like that destroys their creativity.
Sadly it didn't seem to be a fun, usuable, creative madness to me (a la a human on certain drugs), but YMMV.
While trying to iterate on a name for a product, I had it output an internal monologue alongside outputting name ideas.
With each iteration of naming I'd ask it to probe its understanding in of human creativity and list how the usefulness/realism of the internal monologue.
After a few loops the internal monologue started modeling fleeting memories about things like "that time at camp where X happened".
Eventually it was modeling fleeting thoughts like "the warmth of the cup of coffee on the desk" as a source of inspiration, and random observations like "it looks like traffic is picking up outside, getting to yoga will take a while" in the middle of a name generation task
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I made a website predicated on a similar idea after that: https://notionsmith.ai
Technically it's a hallucination generator, but I think you described why I made it (creativity and hallucinations really are similar for both LLMs and humans)
It takes coaxing (and some cherry-picking) to get anything creative out of LLMs -- and some people suspect newer versions are so locked-down to avoid sounding "scary" that they actively avoid saying anything too interesting or philosophical.
The line between "creative output" and "hallucinated madness" is also somewhat blurry (for humans and LLMs alike...)