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Definitely, the one I like the most is the proof via ultrafilters. You only have to state the compactness of a topological space in terms of ultrafilters, which is easily obtained by the definition via open coverings (warning: the equivalence of the definitions is where one uses AC)

X is compact if and only if every ultrafilter is convergent.

Then one observes that

1. any continuous image of an ultrafilter is an ultrafilter (in particular, any projection from a product space)

2. any filter in the product space converges if and only if all its projections converge .

You really only need a few definitions and few natural properties. My test about how nice is a proof is: can I teach it to somebody just while standing in the queue at the canteen, on into subway car?

2 added 102 characters in body

Definitely, the one I like the most is the proof via ultrafilters. You only have to state the compactness of a topological space in terms of ultrafilters, which is easily obtained by the devinition definition via open coverings : (warning: the equivalence of the definitions is where one uses AC)

X is compact if and only if every ultrafilter is convergent.

Then one observes that 1.

1. any projection continuous image of an ultrafilter in the product to a factor is an ultrafilter ; 2. (in particular, any projection from a product space)

2. any filter in the product space converges if and only if all its projections converge .

You really only need a few definitions and few natural properties. My test about how nice is a proof is: can I teach it to somebody just while standing in the queue at the canteen, on into subway car?