There is one good mountain bike permitted trail in Mount Cook National Park, a UNESCO world heritage area. That trail is a rough four wheel drive track that climbs relentlessly for 6km as it gains 300 metres (1000 feet) up to the crest of the lateral moraine of Tasman Glacier. The track is bouldery in places and quite hard going but it you're fit and strong you'll be able to ride all the way.
From the crest of the moraine you get stunning views of the rock covered Tasman Glacer and the lake trapped behind it's terminal moraine. Mount Cook, 3724m, 12248 feet, and other mountains, tower over the glacial valley up head.
Having come this far you should proceed another 4km on foot up to Ball Hut where you'll almost be right under Mount Cook.
The ride back down is a blast on a full suspension bike - you hardly have to pedal all the way.