You only need to download tokens for people who tested positive in the recent past (14 days - assuming you're not infected, any contact before that couldn't have infected you.)
While that's likely still a lot of data in areas with large daily case counts, at least it won't keep growing indefinitely.
BTW it's one token per day per person, to improve privacy. When someone tests positive, their tokens for the last 14 days are published.
When you first install the app you might have to download all keys from the last 14 days, which would be about 15 MB.
16 -> 8 halves those data figures; diminishing returns from there, but it is possible to reduce the data use further (slightly).
With 20 bits/key such filters yield a false positive rate (FPR) of 0.001%. Meaning for 100,000 new cases per day worldwide, you would only need to download 0.24 MiB.
Note that the FPR applies to each lookup. That is, if you have collected say 100 tokens on a given day, the overall probability of a false positive will be 0.01% (assuming independence). With each extra bit per key you can roughly halve that probability. So in practice size won't be an issue.
[1] https://www.cs.cmu.edu/~dga/papers/cuckoo-conext2014.pdf
For people with 200MB & 500MB data caps (which is a surprisingly large number of people), this is a significant amount of their data allowance.
These are likely to be people that are still required to go out into the world and work, so are still using their data instead of wifi, and thus higher risk.
It's easy to forget when you're on 10GB+ data plans for years just how many people are not.