There are no unclimbed mountains in Australia, there might be some unclimbed routes to some of the more remote mountains but the previous answer was completely incorrect, Mount Kosciuszko is climbed 100,000 times a year.
The highest mountain in Australia is Mt Kosciuszko, at 2228m (some sources say 2229m), or 7310 feet. It is situated in the Australian Alps, in southern New South Wales.
Whilst Australia's highest mainland mountain is Kosciuszko, Mawson Peak on the Australian external territory of Heard Island is higher, at 2745 m.
Although the mountain with the highest peak on mainland Australia is generally recognised to be Mt Kosciuszko, there is still some debate about whether explorer Paul de Strzelecki named the highest mountain Kosciuszko or whether Mt Townsend was the higher, as his instrumentation was not completely accurate. There is some slight evidence to suggest that the map-makers assigned the alternate names to the mountains, and that Mt Townsend should in fact be called Kosciuszko and vice versa.
The biggest mountain in South Australia is Mount Woodroffe
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The highest mountain in Italy is Mont Blanc, which has a height of 15,781 feet. The second highest mountain is Lyskamm, with a height of 14,852 feet.
Pico de Orizaba with 5,636 metres (18,490 ft) of height is such mountain.
Mount Elbrus stands at 5,642 meters and is the tallest mountain in Europe.
Queensland is the 2nd largest state of Australia
The highest mountain in North America is Mount McKinley, it has a height of 6 198m
mount kosciuszko is australias highest and biggest
Australia's third highest mountain is Mt Twynam, in the Australian Alps, with a height of 2195m. It trails Mt Kosciuszko (2228m) and Mt Townsend (2209m).
Queensland. The tallest building in Australia is the Q1 Tower on the Gold Coast, with a height of 322 metres.
Mauna Kea is the Worlds biggest mountain.
Aoraki/Mt Cook in New Zealand Altitude 3,754m