There is only one simple requirement for any AI to begin recursive self-improvement: Learning of the theoretical possibility that more powerful or efficient algorithms, preferably with even more brainpower, could achieve the AI's goals or raise its utility levels faster than what it's currently doing.

Going from there to "Let's create a better version of myself because I'm the current most optimal algorithm I know of" isn't such a huge step to make as some people seem to implicitly believe, as long as the AI can infer its own existence or is self-aware in any manner.