Google Contacts (Gmail address book)
They're the ones I've noticed, not sure if anything else is involved.
In Settings > Accounts & Sync, the "Background data" and "Auto-sync" boxes are ticked. In Manage Accounts, my Gmail address is set to sync Contacts, Gmail and Calendar (but not Picasa Web Albums).
I've gone to my Gmail Inbox on the phone, where it indicates, when I "go to labels", "Sync last 4 days" - and showing in the Inbox is a single email from 18 Nov, which has long gone from my Inbox in Gmail on the web browser on my laptop.
... while writing this post I've gone into the Settings in Gmail on the phone, and discovered the "4 days" is something I can alter, and have changed it to "1 day" to see if it makes any difference, which it didn't.
I'm not sure where else to look for settings to make this work. Does anyone have any ideas?