For currency, airline pilots have to do various approaches every so often to remain current or for the plane to be current (RNAV/ILS Cat II/III approaches). So, every now and then when coming to another airport they could request to shoot that approach instead of the one being used. Sometimes it's a non event (no traffic, or the few other aircraft either fit well before or behind them); other times, it could hinder the flow of operations if they're allowed to do full or certain procedures (various circling to land procedures).
Another very common request that falls in this category is to land or depart in the opposite direction of what's being advertised. This in recent years has become more cumbersome for controllers (due to more and new rules implemented), so it's being used less often.
Sometimes when there's little traffic it's fairly easy to accomplish this kind of arrival or departure. Other times, it becomes a nightmare, and unless it's operationally mandated the pilot will be told, unable opposite direction, due to traffic. When it's operationally needed, then it will be told to hold somewhere to wait until there's space to get them out for departure; or vectored around or put in holding if needing to land opposite direction, and they can be accommodated.