Incompatibility: a Google game
Some restrictions that I think are necessary to make the game interesting:
1- The words must be both in the same language (though proper names are allowed and considered to belong to all languages).
2- Both words should individually get at least 10,000 Google hits. This is to prevent tricks like using some obscure Russian name that appears only in a few pages in that language in conjunction with a very common English word like "the".
3- When I tried to start playing, I found that there are a few webpages, such as this one, that appearently try to list all words existing in all languages. I make a special rule to the effect that these pages don't count, so a search that gets only one or some of them is valid as "zero hits".
The best score I got after some time of trying was for the words: pharsalia supersymmetric, which gets 0 hits while the disjunction pharsalia OR supersymmetric gets 4,170,000. I guess that there is just no way of connecting a Latin epic poem and a symmetry relating bosons to fermions. By the way, their individual scores are respectively 963,000 and 3,120,000, which don't add to 4,170,000. I have no idea what Google is doing, but I will accept the results of the combined search with OR as official.
I have just realized that this game has a paradoxical self-defeating character: from now on there will be a Google hit for the "pharsalia supersymmetric" combination, this page! So perhaps a better way of thinking of the game, at least when its results are published on the Web, is not as discovering incompatible words but as creating, with an original act of mind, novel compatible words. I may have been the first person ever to think of Pharsalia and supersymmetry together, but from now on, the combination exists. Wouldn't you like to have that same honour for some pair of words? (Or perhaps you would, in a romantic fashion, prefer to keep some areas of the semantic space forever untouched by mankind?)
So, the challenge is set. Write your own incompatible pairs in the comments (even if they don't beat my score, I'm interested to see what you come up with). I declare the competition open for 5 days; on Tuesday evening, I will establish the winner as the first World Champion of Incompatibility!