Sampling with replacement, expected number of repeats

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I have a bag containing $100$ unique balls. I sample from this bag, with replacement, $100$ times.

From this related question answer I can compute the "expected number of duplicates", or using the Binomial Theorem I can compute the probability that I will see any specific ball drawn twice, or $k$ times. However I can't figure out how to determine:

How many times would I expect the most drawn ball to be drawn from the bag?

I suspect it's something to do with the a sum of the binomial distribution, but as you can tell I'm no mathematician.