The Maharaja of Mysore’s exquisite Anglo-Indian summer palace is full of totally over-the-top princely colonial style. All 19 rooms are gigantic suites, with antique furnishings, teak flourishes, tiled floors, fireplaces and hot tubs. Some are pristine, others need a paint job. You can play billiards, stroll the forest-fringed grounds and dine beneath wood-carved ceilings. The Maharaja himself stays here sometimes.