What badges have their notification suppressed if the user has a 200-rep account somewhere on the Stack Exchange network, i.e. they won't trigger a new item in our achievements dialog?
Silver and gold badges are always considered veteran-worthy, so you'll always be notified of those regardless of reputation or site.
On per-site metas, you're always considered a veteran; on all other sites (including Meta Stack Exchange), you're considered a veteran if you have the network association bonus rights, i.e. you have earned 200 rep somewhere in the past.
The following bronze badges are considered veteran-worthy:
- Nice Answer
- Nice Question
- Popular Question
- Vox Populi
- Tag badges
Any bronze badge not in this list is not veteran-worthy, i.e. won't cause a notification for veteran users.
Back before its retirement, the Tumbleweed badge was also included on the above list.