Jacob, Sean,
Our database shows you currently have no open invitations to join a group. This is why you don't see any invitation button on the Groups tab. If you were invited to a group, maybe there's some other issue that the invitation didn't actually get delivered to your account.
If you were requesting membership to a group, you would not see the invitation button, nor would you be in the group until the administrator of the group approved your membership. After their approval, the group would simply show up on the Groups tab.
Jacob, your user profile is currently set to private. This means your name is not searchable within ArcGIS.com. Thus, another user who may have wanted to invite you to join their group would not have been able to find you in the system, so I'm not sure how they attempted to invite you. I suppose they could have if you had your profile set to public, then recently changed it to private.
In any event, I'd suggest you ask the group owner of the group you want to join to invite you again. (Jacob, make your profile public first.)
Thanks,
Mike