Adjective meaning "argumentative for argument's sake"?

Quarrelsome might be appropriate.

"apt or disposed to quarrel in an often petty manner"

The word you use in your question, argumentative, can be used to express what you want to say. An alternative could be contentious, meaning (for a person) liking to argue.

The type of person you would be is a contrarian, and this word has some currency with Christopher Hitchens. The adjective is contrary, emphasis on the second syllable, as in the nursery rhyme, but this may be mostly BrE.

Eristic, from classical Greek, means to argue with no goal in mind. As the philosopher Gilbert Ryle points out, "the eristic preoccupation with victory displaces any commitment to truth."