One of those interesting things is batting average. For anyone who doesn't know, batting average is essentially just a success rate (number of base hits divided by number of official at-bats) and is conventionally rounded to three decimal places.
Since it's early in the season, no player yet has a very large number of at-bats. As of right now (9:30 am East Coast time on Tuesday April 10th) the player with the most at-bats in the majors is David Freese of the Cardinals, with 24. (Freese has 10 hits in those 24 at-bats, for a batting average of .417. It would be .416666667 if we used more digits, but the convention is to round to the nearest thousandth.)
I claim that nobody will have a batting average of .249 until sometime in May.
One way to approach this topic is just to brute-force everything. The previous paragraph suggests an algorithm: For each n, compute each of the fractions 1/n, 2/n, ..., (n-1)/n as a decimal, and round to three places. For each possible average from .001 to .999, keep track of the first n that gives you that average.
If we do this, we find, for example, that to get an average of .249, you need at least 169 at-bats. The fraction 42/169 rounds to .249 (a more accurate value is 0.24852071) and no fraction with a smaller denominator than 169 rounds to .249.
However, there's more structure here. What are some other averages that require a large number of at-bats? A few examples are .334 (requires at least 287 at-bats), .332 (requires at least 184 at-bats), .251 (requires at least 167 at-bats), .166 (requires at least 145 at-bats), .199 (requires at least 136 at-bats), and .285 (requires at least 123 at-bats).
Basically, the "hardest" averages to achieve are the ones that are very close to fractions with small denominators. Achieving .250 is easy: you can go 1 for 4, or 2 for 8, or 3 for 12, et cetera. But .249 is hard. We know k hits in 4k at-bats won't give you .249, because it'll give you .250. What about k-1 hits in 4k at-bats? Unless k is somewhat large, that's "too far" from .250, and will be less than .249. What about k hits in 4k+1 at-bats? Unless k is somewhat large, that also is "too far" from .250, and will be less than .249.
I wonder how easy it is to state and prove a general result. Suppose batting averages were rounded to 4 or 5 decimal places instead. Can we prove that the "last" batting averages to occur as possibilities (when we steadily increase the number of at-bats) are always numbers like .24999 or .33334? (Actually, I think the very last ones would be .00001 and .99999, which by itself is perhaps not difficult to prove.)