Prop forwards in rugby. You want a grisly 35 year old veteran anchoring at least one side of your scrum.
You want a veteran big on your team in basketball. A guy who knows how to use his body to box out, rebound, post up etc. David Robinson, Olajuwon, Duncan, Mailman, Kareem, Alonzo Mourning all still effective from their mid to late 30s.