There's this film, a classic, starring Steve McQueen and Richard Attenborough, set during the second World War at an Air Force Prisoner of War camp in Germany. The film, if you haven't guessed by now, is called The Great Escape, based on real-life events that happened in Stalag Luft III, Sagan, once a German town, 100 miles south-east of Berlin.
In the film there is a Flight Lieutenant pilot called Robert Hendley, an American in the RAF, whose nickname is "the scrounger"
who finds what the others need, from a camera to clothes and identity
cards
Interestingly, the actor who portrayed this role, the gorgeous-looking James Garner, had been a soldier in the Korean war and he too had been a scrounger during that time. Unfortunately, today the term has very negative connotations especially in BrEng where it is synonymous with moocher, sponger and a freeloader but according to War Slang. American Fighting Words and Phrases Since the Civil War by Paul Dickson, its meaning was slightly more noble.
Scrounge. To appropriate; to misappropriate. In Behind the Barrage,
published shortly after the war, British writer G. Goodchild
discusses the term:
In the category of "odd jobs" came "scrounging." "Scrounging" is
eloquent armyese; it covers pilfering, commandeering, "pinching,"
and many other familiar terms. You may scrounge for rations, kit,
pay, or leave. Signalers are experts at it, and they usually
scrounge for wire. Scrounging for wire is legitimized by the War
Office, and called by the gentler name of "salving."
and further on
scrounger. One adept at acquiring food and other goods.
Oxford Dictionaries says scrounger in AmEng is
A cleverly resourceful person who finds and procures items for a
specific purpose
So, it kinda fits if the character is American or is given the nickname of Hendly; however, the author would have to give a brief justification for this name.
Best Answer
As implied by the question itself, the standard term for the behaviour itself is one-upmanship.
Although alphadictionary.com is prepared to accept one-upman as a word in itself, to be honest I think they're in the minority on that one. In the absence of a convenient single-word derivation from the standard term, people usually just say it using more words - for example...
Most of us indulge in one-upmanship from time to time - it's part of normal social interaction. It only becomes tiresome with those few people who do it excessively. The most common single-word adjective I can think of for such a person is competitive. Which has lots of other shades of meaning, but when used in the context of how someone interacts socially, I think most people will know exactly what you mean.
user653's comment to @Barrie's answer reminds me of my favourite example from Dilbert...