Polarized sunglasses are a "must have" accessory for any fly angler to better spot fish and minimize eyestrain and fatigue. Sunglasses also protect your eyes from rogue casts.
Clippers or nippers, for cutting tippet, the tail ends of knots, etc.
Zinger, for holding your nippers or hemostat.
Hemostat (a clamp or forceps used by anglers) for removing flies from the mouths of trout, and for pinching down the barb on the hook.
Nail knot tool, for helping you quickly tie leader or backing to fly line.
Fly floatant, for helping the fly stay afloat on top of the water. This is powder or liquid that you apply to your dry fly to keep it from being saturated in water.
Extra packages of tapered leaders or tippet spools to replace knotted or worn leaders.