Tamanrasset itself is a functional desert town without major sights of its own. The attraction is what surrounds it: the volcanic peaks and valleys of the Hoggar Mountains, best explored on multi-day 4x4 expeditions with a local guide.
The unmissable destination is Assekrem, an 80-kilometre journey on rough tracks to a high mountain pass at 2,780 metres where Charles de Foucauld built his hermitage in 1911. Arrive late afternoon, sleep near the pass, and wake before dawn for the Saharan sunrise — one of the most extraordinary sights in North Africa. Serious walkers can continue to Mount Tahat, Algeria's highest peak at 2,908 metres, most easily approached from the Assekrem area.
Further afield and requiring a dedicated expedition, the Amguid Crater is a near-perfect meteorite impact crater about 250 kilometres north of town — remote even by Saharan standards, and visited by very few people.
In town, the market is worth a wander for Tuareg silver jewellery, leather goods, and the distinctive blue and white clothing of the Kel Ahaggar Tuareg. All excursions outside Tamanrasset require a registered guide and permits — arrange these through one of the tour agencies in town before you arrive.