I have an API that is somewhat popular (10,000+ requests/day). After 10 requests per day from an IP address I return a message telling the user they need to cough of some cash i
Block the entire subnet of the abuser to solve the problem temporarily. These types of users will appear from other countries as well so your best bet may be to require a registration and an API key to use the API.
If you still want to block based on IP rather than API key, check how large the abusing subnet is using whois (or BGP) and block the entire IP range.