Hiking boots, good sturdy boots with sufficient ankle support for walking on rough trails. They should have a robust sole and preferably waterproof.
Warm jacket, fleece, primaloft or similar Rain jacket with a hood, preferable breathable fabric such as Gore tex or similar Rain trousers, preferable breathable fabric such as Gore tex or similar Gloves, wool or synthetic, preferable waterproof.
Hiking socks Warm hat and/or balaclava Backpack for daily use Sleeping bag and travel pillow.
A light sleeping with no specific temperature rating will be sufficient Liner if you rent the sleeping bag
Towel, Tooth brush and personal toiletries
Sunglasses/sunscreen (high protection, the sun in Iceland can be really strong)
Water bottle Small personal first aid kit
Pain killers such as aspirin or ibuprofen
Hiking poles
Gaiters
Camera, batteries, memory card
Book Headlamp for trips in late season, after august 10th. Earplugs/Sleep mask
