Is there a maximum number of API calls that can be made in one day or per second?

And if not, does Backendless throttle service at a certain point? Also would I be able to pay for more?

If backendless is hosting your app, then its 50 API calls/second for the free plan. See https://backendless.com/pricing/backendless-cloud-functional-limits/ for more on limits and extension costs.

The free limit is 50 req/sec prorated by a minute, which means after the first 3000 requests in any given minute the system will reject incoming requests unless the limit is increased with a Function Pack.