Interesting responses. Thank you all.
Chance is a fascinating game design element. I find it works best when it is something that causes gameplay to change from one play to the next, rather than something that determines who wins and who loses. For instance, in the boardgame "The Settlers of Catan", the initial board setup is random, so it's never the same from one game to the next. But on each individual play, all players must play on that same board. So this random element plays no part in determining the winner.
By contrast, in "Monopoly", the die rolls (particularly during the endgame) are the sole arbiter of who wins. Did you land on your opponent's most valuable property? Did they land on yours? In the end, nothing else matters.
Then there is the middle ground; games like "Euchre", for example. If you and your partner get good cards, you might win if you play them correctly. Play them incorrectly and you'll lose for sure. But if your opponents have better cards than you, superior play won't help a damn unless they make some very dumb mistakes.
For many gamers (including, apparently, most of the people on this board), this middle ground is the ideal. Not for me, though. If I get wiped out of a game because all of my neighbours got lucky and had their planets magically transform into class 17 juggernauts, I feel like I've been wasting my time struggling through an unwinnable game. The response from most CG players would appear to be "suck it up and quit whining", but i'd rather lose on account of being out-thought and out-played by my opponents than on account of unlucky (virtual) dice-rolls.
Matter of taste, I suppose.