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 via open coverings:
X is compact if and only if every ultrafilter is convergent.
Then one observes that
- any projection of an ultrafilter in the product to a factor is an ultrafilter;
- 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?