WWTBAM Bored

Since my math skills are lacking, and I know there are some here with great skills, I figured I'd ask during the off-season.

Every once in a while, I'm crazy enough to play the Powerball with some people at work (I know, an investment for people who can't do math).

In our 25 games, we only had one number once on all the games! Of 125 numbers on our tickets, plus the 25 Powerball numbers selected (OK, some were repeats), we had a grand total of one number correct (that number did not repeat).

There are 5 white balls of 59 drawn; and one Powerball of 35 red balls drawn.

What are the odds of our futility of only getting one white ball correct in the 25 games we played?

Your expectation value with 125 balls and 59 different numbers would be about 125/59 = 2.1 correct, so 1 isn't very surprising.
Here's what I get with a full calculation:
0 balls correct: 10.9% chance
1 ball correct: 25.3%
2 balls correct: 28.1%
3 balls correct: 19.9%
4 balls correct: 10.1%
5 balls correct: 3.9%
6 balls correct: 1.2%
7 balls correct: 0.3%
8 or more balls correct: <0.3%

Should I look at the tickets and see how many different numbers (since I'm sure some numbers repeated) we did have in the 25 games played?

We've played about 80 times, and although we've had some bad times, this was by far the worst. Not even getting a Powerball correct (which isn't in your calculations).

I would just think if there isn't a misunderstanding of what I'm asking, it would mean this should happen about one every four times we played, but it only happened once in 80.

As long as numbers can't repeat within a single game, repeat numbers don't affect the odds because each game is independent of the others.
For the powerball by itself, your expectation value in 25 balls with 35 numbers is 25/35 = 0.71 correct, so 0 isn't surprising. Odds for the powerball in 25 tries are:
0 correct: 48.4%
1 correct: 35.6%
2 correct: 12.6%
3 correct: 2.8%
4 correct: 0.5%
5 or more correct: <0.2%

Thinking about it a little more, I realize I mistakenly calculated the odds above as if the position in the draw mattered for each of the 5 balls drawn. Since it doesn't matter, that increases the expectation value by 5, so it's about 10.5 instead of 2.1 in 125 balls drawn. So you are right that only 1 correct is surprisingly low. I calculate the odds of 1 correct as only 0.013%.

I think it might be even less considering we didn't get the power ball correct in 25 tries as well.

I just noticed your location. Love it. It's mad, mad, mad, mad.

To clear up any misunderstanding of what happened, I just checked the tickets from this. It was the February 7th drawing. We did a little better than I thought, but this still has to be some real long odds:

*Of the 59 white balls, we had 53 of them in our 25 games played.

*The numbers drawn were 5, 10, 21, 34, and 58.

*We had 21 on two games; and 34 on one game.

*Of the 35 red balls, we had 19 of them in our 25 games played.

*The number drawn was 33--we didn't have that number at all.