Here’s how it works:
1. A “starter bracelet”—one link and gold-filled chain—is purchased.
2. For each gift occasion, a new link is purchased and permanently installed.
3. Once completed, the item looks like an expensive tennis bracelet.
Start-to-Finish LLC is a manufacturer of diamond link bracelets that are sold by fine retail jewelers. Here is a picture of a Start-to-Finish(TM) single-link bracelet style.

The other product that comes to mind also sold in jewelry stores is Add-a-Link(TM).