How long does it take for a frog egg to become an adult frog?

This all depends on the species. Some frogs, especially desert-dwelling species where the tadpole's pools may only last a few days, have completed metamorphosis in under a week. Some tadpoles take three or four years (Rana species in America in mind but there are others). For the Common Green Tree Frog/White's Tree Frog (Litoria caerulea) which is my personal favourite species:

36 hours for the eggs to 'hatch'. The black dots inside the jelly develop into tadpoles which wriggle free of the gel.

4 weeks - back legs start growing.

10 weeks - back legs finished growing.

12 weeks - first front leg

1 day after first front leg - second front leg. Within a few days the tadpole will take on a frog's shape, stop eating and start to reabsorb its tail. It will become a much weaker swimmer and will need to be able to get out of the water. ^ This is the development of a relatively fast-growing Australian species that takes about 3-4 months to complete development under OPTIMUM CONDITIONS - this is, as much food as the tadpoles can eat, clean water changed daily, water kept warm but not too hot. These conditions are seldom perfect in the wild, resulting in the death of well over 60% of the clutch before metamorphosis and much slower growth in most instances.