What is this number 1 on my Google Chrome icon (Mac)?

The number represents active downloads...

Those if you have a transfer pending (or paused, or perhaps corrupted), it may not show up in the download list, yet it's still "active".

Click the Chrome icon, and choose "quit", if you have any pending/active/paused downloads, it will offer you a chance to cancel or continue them. Cancel them, exit Chrome, and the number overlay should go away.

For more info/discussion perhaps check out this Google Groups posting.