How big can the Algorand Blockchain get before problems start occurring?

I see.
Algorand Inc/Foundation itself does not provide such services.
However, partners/third parties provide such services. A list is there:

These API services are free with rate-limiting and no SLA is provided.

This is similar to what Stellar Horizon seems to propose.
I see they also have rate limiting:

And I don’t see any SLA / customer service (but I did not look very long, so I may be wrong).

I’ve never developed on Stellar. But I would guess that if I had to use a Stellar API on a big project, I would want to pay for an API tier with high SLA, 24/7 customer support, and lower rate limitations.
Similarly as what I strongly recommend on Algorand.