Current non skill/quest requirements that I can think of :
-Temple trekking (Morytania)
-Chompy kills (Ardougne or Western lands) - 1k kills.
-Black knight kills (Falador) - Upto 1.3k kills.
-Rank 5 all BA (Fremennik)
-Museum Kudos (Varrock) - Upto 153 kudos.
-Unlock Bones to Peaches (Desert) - Thanks to Mini Finbarr
-Mage Arena capes / spells unlocked (Wilderness) - Thanks to Mini Finbarr
-Barbarian training and barbarian notes (Fremennik) - Thanks to Mini Finbarr
-Unlock the highest level of Ali Morrisanes rune shop (Desert) - Thanks to Mini Finbarr
-Complete the rat poll and cat training medal (Varrock?) - Thanks to Mini Finbarr
-All hot air balloon routes unlocked (Unknown)
-Pick a Cave Nightshade (Ardougne) - Requires Skavid Map + Watchtower quest