Fastidious can have the sort of connotation
I think you're looking for.
fas·tid·i·ous ( adj .)
Possessing or displaying careful, meticulous attention to detail.
Difficult to please; exacting.
Excessively scrupulous or sensitive, especially in matters of taste or propriety.
"He was the sort of guy who might ring you at 4 in the morning to repay sixpence; the chap was fastidious about his debts."
english.stackexchange.com/users/4763/Uticensis
Perhaps punctilious would be the word you are looking for?
english.stackexchange.com/users/305/Brian Hooper
I would call such a person overly or obsessively proud , or possibly haughty .
But it would depend on the culture.
Such a practice is considered normal in Japan, for example, where the concept of on ( face ) requires it.
english.stackexchange.com/users/2303/Robusto
I think the word stickler fits very well here, in the sense of a person who's obsessive about making sure customs and rules are followed to the letter.
If I were to use it in a sentence, it would go something like this:
He was a stickler for making sure that even the smallest and most insignificant debt was repaid in full.
It carries a connotation of following a rule to the point of being slightly annoying about it.
english.stackexchange.com/users/72089/Oskar
I'd go for "stubbornly independent, even in the face of generosity".
english.stackexchange.com/users/3328/Zoot
I would say this person suffers from an aversion to generosity .
That's not an adjective, but I think the phenomenon may be too complex to boil down to a single word (cultural factors?
fear of indebtedness?
fear of not being able to return a kindness?
feelings of inferiority?
etc).
Maybe generosity averse ? :)
english.stackexchange.com/users/4859/adj7388
I would call him a nitpicker, after revieing this
I'm not sure whether it would be approriate: nitpicking
english.stackexchange.com/users/519/stacker
How about compulsively reciprocal ?
english.stackexchange.com/users/9727/Bryce 
Someone who is " anal retentive " is a good fit and conveys the speaker's annoyance quite effectively.
Wikipedia describes it thus: The term anal retentive (also anally retentive), commonly abbreviated to anal, is used to describe a person who pays such attention to detail that the obsession becomes an annoyance to others, potentially to the detriment of the anal-retentive person. .
english.stackexchange.com/users/44619/Mari-Lou A
Many Danes are familiar with a concept that also seems to be in play in the case of the OP's obsessive debt-repayer.
They call it millimeterdemokrati ("millimetre democracy").
It is typified by phenomena such as the scrupulous analysis of a restaurant bill to calculate who owes how much, the maintaining of a log in order to track whose turn it is to wipe down the fridge door or rinse the tea cups, and the cutting of a cake into slices of precisely equal size (with the help of kitchen scales, in the worst case).
You could say that individuals afflicted with this impulse are applying an ethos of egalitarian reciprocity/responsibility taken to an extreme degree.
(I hasten to add that most Danes are not quite this silly.)
Looking over the other suggested names for this attitude, I don't think any of them quite capture the nitpicking quality of millimetre democracy at its finest.
english.stackexchange.com/users/71980/Erik Kowal
Let's say you paid for a friend's bus ride, and he insists on paying you back rather than just let it be and stop worrying about it. You even find it rude on his part that he won't just forget about it. Is there an adjective or a phrase for such kind or a person, or for such kind of a behavior?
What would you (negatively) call a person who insists on repaying even a small amount of money?
english.stackexchange.com
<single-word-requests><adjectives><phrase-requests><pejorative-language>
english.stackexchange.com/questions/13027
english.stackexchange.com/users/5159/Ilya Kogan
There are number of terms or phrases that might describe such a person or behaviour. These include "difficult to please", "excessively scrupulous", "punctilious" or "anally retentive". You could also say that this person suffers from an aversion to generosity. There are numerous adjectives that can apply including "careful", "meticulous" meaning someone who pays extra attention to detail. An alternative is "punctilious" or "stickler" which has a slight annoying connotation. Another option is "stubbornly independent" or "obsessively haughty". However, the trait described could be a cultural trait and a fear of becoming indebted in some cases; one suggestion is "adverse to generosity".
There are number of terms or phrases that might describe such a person or behaviour. These include "difficult to please", "excessively scrupulous", "punctilious" or "anally retentive". You could also say that this person suffers from an aversion to generosity. Fastidious, punctilious, nitpicking or compulsively reciprocal could carry this meaning. Other descriptions like being a stickler or 'stubbornly independent, even in the face of generosity' could apply in some contexts.
A person who insists on repaying even a small amount of money could be called a stickler, a person who's obsessive about making sure customs and rules are followed to the letter. Or stubbornly independent, even in the face of generosity. Or obsessively proud, possibly haughty. Punctilious, excessively scrupulous or sensitive, especially in matters of taste or propriety.      There is no agreed term for this concept, but some potentially acceptable terms include "exacting", "haughty", "stubbornly independent" or "compuslively reciprocal".
It can mean paying extra attention or care to detail or be difficult to please. Possessing or displaying careful, meticulous attention to detail Maybe "puncitlious" is the word needed. Obsessively proud or haughty person. "stickler" is another term, someone annoying regarding detail. "stubbornly independent" is another alternative "averse to generosity" "nitpicking" "compulsively reciprocal"  "anal retentive"
Fastidious, punctilious, nitpicking or compulsively reciprocal could carry this meaning. Other descriptions like being a stickler or 'stubbornly independent, even in the face of generosity' could apply in some contexts.
There is no agreed term for this concept, but some potentially acceptable terms include "exacting", "haughty", "stubbornly independent" or "compuslively reciprocal".