From Steinbeck's Cannery Row,
Doc was almost supernaturally successful with a series of lady visitors. He didn’t half try.
Does it mean that Doc did not do anything to win hearts of these ladies?
I have to disagree with FumbleFingers. In the Cannery Row example it means Doc didn't seem to put any effort into it at all, but walked away with remarkable success. Think of it as
I see FF has edited his post to add something about this before the deadline, so I'll just emphasize that this is the meaning, and it's not a statement that Doc was trying hard.
To not half [do something] is a common idiom meaning to do it completely/wholeheartedly.
Having said that, particularly in older texts, the construction can have more or less the opposite meaning (effectively, "I don't half like = I don't very much like"). But in Steinbeck's usage, I think we can safely say Doc puts a lot of effort into ensuring success with the ladies.
EDIT: I don't really want to be forced into having an opinion on which of the two possible meanings applies in this specific context. I'd expect it to be obvious to anyone who'd been reading the entire text up to that point - but that doesn't include me, and ELU isn't a Lit. Crit site.
By way of showing that in general the idiomatic meaning I gave is currently the more common, consider these 15 instances of "I won't half give" - which in every single case clearly means I will definitely give.