When selling downloadable files, is there an existing way to track the number of times each user has downloaded a file after purchasing?
Store doesn't support this natively.
There's nothing special about the way Store handles downloads, except that it provides a neat way to hide the real URL to a file and require a purchased order to visit the link. Since the files are just stored in regular EE channel fields, there is certainly no requirement that you use Store's download tag.
Therefore, if you want more advanced features or reporting, as Peter mentioned your best bet is to look to third party add-ons. Link Vault is definitely the most popular that I'm aware of.
I think, its late reply.
I developed a module "Member Downloads". One can see the documentation from here: http://www.sidd3.com/member-downloads-tracking-members-downloads/. It seems something suitable for you.
Soon, it will be available at devot:ee to purchase.