How to get free programmatic access to citations counts for a given paper? What databases allow free, programmatic access to cited-by counts for papers?


Google Scholar doesn't have an API, and they block crawling attempts.
SCOPUS only seems to allows you to retrieve a branded, watermarked image of the number.
If you are an R user, you can use the scholar package. This package allows you to analyze data from Google Scholar and obtain citations, publication list and even perform predictions of the h-index. An example R code:

library(scholar)
id = "xJaxiEEAAAAJ" # Isaac Newton's id
cit=get_citation_history(id)
barplot(cit[,2],names.arg = cit[,1])