You could use the Bitbucket API, the endpoint below returns the 15 most recent changesets, but also returns a count which is the total number of changesets.
https://bitbucket.org/api/1.0/repositories/{accountname}/{repo_slug}/changesets/
Check out the REST API Console (http://restbrowser.bitbucket.org/) to play with the API.
You can also simply clone the repository and run the following command in GitBash or OSX terminal:
git log --oneline --all | wc -l
This will output every commit in the repository, one per line, then count all the lines.
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