How often was the most frequent coupon chosen?

Each coupon appears $\log n$ times on average.

If I treat the number for a single coupon as a binomial, the variance is also asymptotically $\log n$.

Now treat each of the counts as a normal variable $\mathcal N(\log n,\log n) $.

Expectation of the maximum of gaussian random variables shows the maximum of $n$ independent standard normal random variables to be around $\sqrt{2\log n}$, so the final answer would be $$\mu+\sigma \sqrt{2\log n}=\log n+\sqrt{\log n}\sqrt{2\log n}=(1+\sqrt2)\log n$$

Edit:

On the other hand, here is an argument that the limit is $e\log n$.

Treat each coupon count as an independent Poisson variable with parameter $\lambda=\log n$. The tail of the distribution is given here as $$e^{-\lambda}\frac{(e\lambda))^x}{x^x}=\frac1n\left(\frac{e\log n}x\right)^x$$ The chance that at least one coupon is inside this tail would be around $n$ times that. So the probability moves from near-zero to near-1 when $x$ is near $e\log n$.

Here is a simple simulation in Python 3:

import random

def completeCollection(n):
    #collects random "coupons"
    #in range 0,1,...,n-1
    #until each is encountered at least once
    #returns the list of counts

    counts = [0]*n
    collected = set()
    while len(collected) < n:
        coupon = random.randint(0,n-1)
        counts[coupon] += 1
        collected.add(coupon)
    return counts

def expectedMax(n,trials):
    #estimates the number of times
    #the most collected coupon is
    #collected while collecting n coupons
    count = 0
    for i in range(trials):
        count += max(completeCollection(n))
    return count/trials

I evaluated [expectedMax(n,1000) for n in range(1,101)] and obtained:

[1.0, 1.985, 2.79, 3.403, 4.052, 4.575, 4.784, 5.145, 5.45, 5.672, 6.008, 6.198, 6.515, 6.508, 6.765, 7.006, 7.074, 7.207, 7.466, 7.416, 7.534, 7.711, 7.812, 7.992, 8.049, 8.012, 8.268, 8.467, 8.408, 8.467, 8.604, 8.54, 8.779, 8.666, 8.804, 9.121, 9.076, 9.033, 9.179, 9.289, 9.344, 9.33, 9.479, 9.456, 9.601, 9.613, 9.644, 9.836, 9.693, 9.82, 9.886, 9.944, 10.044, 10.124, 10.161, 10.113, 10.039, 10.273, 10.334, 10.345, 10.317, 10.454, 10.519, 10.483, 10.491, 10.496, 10.617, 10.593, 10.719, 10.859, 10.885, 10.782, 10.858, 10.841, 10.87, 10.804, 11.005, 11.005, 10.993, 11.105, 11.092, 11.121, 11.106, 11.159, 11.198, 11.209, 11.291, 11.393, 11.444, 11.428, 11.395, 11.584, 11.533, 11.511, 11.545, 11.559, 11.601, 11.585, 11.61,11.617]

Plotted this looks like:

which looks somewhat logarithmic. It also looks somewhat like a square root, but I evaluated expectedMax(1000,1000) and got 17.835, which suggests that the growth rate is slower than a square root.