Known as the Queen of Ranthambore, Machli was described as one of the world's most photographed tigers. She had distinctive fish-shaped markings on the left side of her face.
She featured in many documentaries and amazed viewers when her fight with a 14-foot crocodile was caught on film.
She was one of the biggest draws for thousands of tourists visiting Ranthambore National Park,
India has more than half the world's tigers, at 2,226 in the latest estimate.
Machli was found starving and lying on her side near the park's boundary in northern Rajasthan state. She had not eaten in days, officials said.
"We were trying to provide her treatment but she died," Ranthambore tiger project director Yogesh Kumar Sahu told AFP news agency.  "It was a natural death linked to her age."
Machli had slowed down in recent years and lost almost all of her teeth, according to the park's website.
Her death has made headlines in India, and fans have taken to social media to pay their respects to the Queen of Ranthambore.
She was named Machli, the Hindi word for fish, because of the markings on her face.
Natural World BBC photo credit: MACHLI - TIGER QUEEN/Natural World/Mike Birkhead Associates/BBC
India's most famous tiger, 19-year-old Machli, has died, officials say.