For me, intuition-wise, I'd order it "undefined, 1, 0".

There are 3 cases for 1 and one case for 0 that immediately spring to my mind when considering the problem:

    0) Limit of 0^x, as x approaches 0 (from above).
    1a) Limit of x^0 as x approaches 0 (from either direction).
    1b) Limit of x^x as x approaches 0 (from above).
    1c) "What did you multiply by 3 once, to get 3^1?  So, multiplying 1 by zero, zero times..."

Limits here, simplified to intuition level, being "what would you need to fill that hole in the graph?" The fact that these disagree would be why I'd assume undefined, but the case for 1 seems stronger (to me).

The limit in 1b is 1 from below as well, right? I'm not sure how limits work with complex numbers, but the imaginary part of x^x approaches zero as x approaches zero from below, so can we say that the limit of x^x as x approaches zero from below is also zero?

Yes, but getting there steps out of the realm of "intuition" for me.

Yes, perhaps "intuition" isn't the best word. Formal limits certainly aren't "intuitive" to me, at least by one definition of the word. I suppose I used "intuitive" to mean "according to my mathematical understanding, ignoring the mathematics explicitly dealing with 0^0."

Sure. I was limiting it to things I could do in my head in tens of seconds.

0^2 and 0^1 may be zero, but 0^-1 and 0^-2 are infinite. So I don't think that intuition leads to 0^0=0. If anything 0^0=1 preserves the symmetry.