How likely / believable is it for a lawful neutral character to worship a chaotic evil deity?
I guess you mean a character who doesn't get any kind of benefits from it, not a cleric or similar that can only be 1 step far from the deity's alignment.
I think it might be possible as long as the character was somehow mistaken about what the deity's real goals and predicaments are. Maybe a god of carnage and destruction could be mistaken for a god of fair vengeance and war, but that could only work with the character being somewhat ignorant of all the credo of that god.
Normally I don't see how a character could be interested on worshipping a god that represents something so different from what the character believes.
Also, if the character somehow realizes what the real predicaments of the god are the logic thing would be that he had a crisis of faith.
In Dragonlance, all sea people usually pay hommage to Zeboim as she is the goddess of the sea and they fear her. Some even pray to her disregarding of their alignment. But they do it out of fear, because they are afraid of her fury and power, not because they like her or believe on her credo. That's another option.
Depends... a man bent on the destruction of the world COULD worship Rovagug... but then that guy would likely be evil
Or a lawful fighter could easily worship Gorum...
Generally, it takes a great affinity between the character's and the deity's ideals to worship a deity, even more so when your alignment doesn't match at all... t*you better have a very good story to back it up.
I could see a chaotic evil god tricking a lawful neutral person into becoming their servant but they'd need a reason to go out of their way for said person.
Seems like something a malicious trickster god would enjoy though. Otherwise I don't really see why it would happen? Do you have an idea in mind.
Well I was looking at options for a Monk, and thought to myself "Gee wizz, Thamir Gixx's always threatening trait sure is nifty."
No plans for actually taking it, it just got me thinking about alignments and deities, and thought i'd get people's opinions on how far is too far to stray from your deity's alignment.
Weird enough to need a good backstory to justify it, I think. And weird enough to make a very interesting backstory if you work it well around it.
Is a LE monk an option? Beside that, there is at least one archetype that removes the alignment restriction: Martial artist.
While it's possible to make up a backstory for a LN character worshipping a CE deity, depending on the campaign that might not be enough. During the sessions, you might encounter intense conflicts between both alignments, up to the point where you have to decide. So I'd make sure the GM is at least neutral about this idea, and doesn't actively work against a character concept they consider 'unnatural'.
In a Rise of the Runelords campaign I played a lawful good ranger/monk gnome who worshipped the chaotic good god Desna. He was lawful because he believed in community and self-discipline, not because he believed the world should be orderly, so his attitude had no conflict with Desna. His chaotic good family in Sanos Forest in Varisia worshipped Desna, and he liked travel, so he kept to the family religion.
More recently, in the Iron Gods campaign I run, I had to fill out some encounters in the slums of the city of Starfall. I created a combined temple of chaotic evil Thamir Gixx and neutral good Chaldira Zuzaristan. The temple was run by a pair of halfling siblings, the sister was the cleric of Thamir Gixx and the brother was the cleric of Chaldira Zuzaristan, and served the halfling ghetto in whichever way was needed.
In a complex setting, alignment and worship don't have to align.