What is the highest mountain entirely within India?

The highest mountain entirely within India is Nanda Devi.

Nanda Devi reaches 7,816 metres and stands in Uttarakhand's Garhwal Himalaya. It is India's second-highest mountain overall, but it is the country's highest summit located wholly within Indian territory. Kangchenjunga is higher, yet its summit lies on the India–Nepal border, so it does not satisfy the question's “entirely within” qualification.

Nanda Devi is a two-peaked massif. The higher western summit is surrounded by a dramatic ring of mountains enclosing the Nanda Devi Sanctuary, a high-altitude basin protected within Nanda Devi National Park. The mountain is also deeply revered in local Hindu tradition as a manifestation of the goddess Nanda, often associated with the daughter of the Himalaya.

Noel Odell and Bill Tilman made the first confirmed ascent on 29 August 1936. The mountain's remoteness, steep terrain and conservation status make it far less accessible than many famous Himalayan peaks. Kamet and Trisul are important Uttarakhand summits, but neither is higher than Nanda Devi.