At least 15m people are expected on the first day alone. Officials project about 120m visitors over 49 days.
Hindus believe bathing at the rivers will cleanse their sins and bring salvation.
The holy men were among the first procession to arrive early Tuesday.
The BBC's Geeta Pandey, who is at the scene, said the men, known as sadhus - or ascetics - smeared ash on their bodies as they came out of the water and chanted "Har Har Gange", or "Mother Ganges", and danced while posing for photographers.
The Naga sadhus are the biggest draw of the festival - held in the northern city of Allahabad, recently renamed Prayagraj - and can be counted on to arrive in massive colourful processions.
At the last Kumbh in 2013, female ascetics were allowed to bathe at the confluence of the rivers - known as the Sangam - for the first time. A few hundred transgender people already bathed on Tuesday morning.
More than a million foreign pilgrims will also take part in the festival, senior administration official Rajeev Rai told the BBC.
He and other organisers have been preparing for more than a year for the event, which dwarfs the annual Hajj pilgrimage to Islam's holiest sites in Saudi Arabia.
"Last minute preparations are on. All the religious sects have been allocated time for their processions and bathing rituals," Mr Rai said. "We have devised a traffic plan to ensure there's no overcrowding. The mela area is open only to pedestrians,"
The mela (meaning "fair" in Hindi) has been held in Allahabad for centuries now, but it has grown into a mega event in the past two decades.
This year the gathering will be particularly huge and many believe India's Hindu nationalist government has organised it with an eye on key general elections due in the summer.
Massive billboards of Prime Minister Narendra Modi dot Allahabad city and the mela ground. Huge cardboard cut-outs have been placed strategically at the bathing areas.
A temporary tent city, spread over 32 sq km (12 sq miles) has been set up to accommodate the masses, complete with hundreds of kilometres of new roads. Hospitals, banks and fire services have been set up just for the festival, along with 120,000 toilets.
Hundreds of new train services are running to and from Allahabad to tackle the rush of pilgrims and more than 30,000 police and paramilitaries have been deployed to provide security and manage the crowds.
In the run up to the festival, religious sects have held daily processions marked by much pomp and show.
At one such procession on Sunday night, there were elephants, camels and horses. Brass bands and drummers played, as religious leaders sitting atop several vehicles threw marigold flowers to thousands of devotees.
On Monday - a day before the official start of the festival - tens of thousands of pilgrims bathed at the Sangam. Some then lit clay lamps and floated them along with flowers in the Ganges.
The atmosphere at the mela is festive, and the authorities have announced a calendar of music and dance performances. But there's plenty of impromptu entertainment taking place by the roadside, with children performing rope tricks and shows by drummers and ballad singers.
Most pilgrims, however, say they are here to "answer the call of Mother Ganges".
"We believe that bathing here will destroy our sins," farmer Pramod Sharma said.
"The waters here have regenerative properties. Bathing here can cure your ailments. It also removes obstacles from your way," Shahbji Raja said.