For the standard Salesforce approval processes, who else can approve a request besides the designated approver?
If you've set the approval to "anybody from the Queue" make sure you're OK with who's in the Queue exactly. Easiest is to click "view all members". Roughly speaking expect that all managers of your queue members (all the way to CEO or whatever is your top role) will have access too.
I honestly don't know if it also works when an User is set as the actual approver (can his Manager do it? can somebody higher in the hierarchy do it?)
Edit: Ralph has tested this thoroughly and while higher Roles do become indirect queue members - this fact alone doesn't mean they can approve. Explicit access must be given or "modify all" permission for the object.
Profiles / Permission sets
As pointed out by Jesse - anybody with "modify all data".
Last but not least
User's own delegated approver, if anybody was set. That's for "what to do when I'm on holidays". Delegated approvers don't get notifications (unless they wish to do so, they can set it on their user record) and don't see the requests on home page. As far as I know if hey don't want spam - easiest for them is to periodically check reports.
Sysadmins (actually "modify all data" again) can view Administrative -> All pending approval requests Delegates can view Activity -> My delegated approval requests.
System Administrators can approve the records as well. I believe it is based on the Modify All permission on the object with the approval process.
A delegated user can also approve or reject .he cannot reassign but he can approve or reject