While the odds for each individual sweepstakes do not change no matter how many times you enter, the odds of you winning actually increase very slightly with each sweepstakes you enter. It's been a while since I did the math, but I was a tutor for combinatorics (looking at the odds of discrete events occurring) in college, so I'll try to explain.
To start with an easy example, look at a deck of cards. The odds of drawing the Ace of Spades is 1 in 52 (assuming no Jokers). However, the odds of not drawing the Ace of Spades is 51 in 52. Let's say you repeat the activity 20 times - each time replacing the card that you drew, and shuffling the deck -- making each draw a separate, discrete event.
The odds of any single one of those draws resulting in the Ace of Spades being drawn is still 1 in 52.
However, the odds of not a single draw resulting in the Aces of Spades being drawn is (51/52)^20 or 67.8%. So the odds that one of those 20 draws result in the Ace of Spades being drawn is (100-67.8) or 32.2%. The odds still aren't good, but you've improved your overall odds by continuing the activity.
This can expanded to sweepstakes -- say you enter 1000 sweepstakes each year, and the odds for winning the grand prize of each of them is 1 in 1,000,000 (You could use the actual odds for each, but 1:1,000,000 makes for easier math). The odds of you winning after entering just one sweepstakes is 0.0001% After one year, the odds of you not winning a grand prize is (999,999/1,000,000)^1000 or 99.9%. So the odds that you would have won a grand prize at least once have improved to 0.1%.
Hopefully that wasn't too confusing. It's probably important to note a few things:
1) The odds of some (many?) of the sweepstakes will be considerably better than 1:1,000,000 depending on the prize, sponsor, and length of the sweepstakes, which push the odds further in your favor. The odds are much better if you count in to win any prize, rather than just the grand prizes.
2) Even in my sweepstakes and card drawing examples above, we're dealing with pretty small samples sizes, so the results can be skewed.
3) You can't win if you don't enter.