What word is appropriate for someone who tries to one up everything you say? For example:
Me: "You are a great friend"
Person: "Well, you are my best friend"
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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...
He always has to one-up everything you say. (lots of written instances in that link)
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...