Not if you have a Google account, they don't.
> If you have Web History enabled, this data may also be stored in your Google Account until you delete the record of your search.
> When you create a Google Account, Google Web History is automatically turned on.[0]
So for everyone who has a Google account, and who hasn't taken specific steps to disable "web history" (read: 99.9% of users) they're storing that data indefinitely.