**Visa information can change at short notice**
With the Mongol Rally finishing in Ulan-Ude, Russia rather than Ulaanbaatar, Mongolia, many teams will need to have a double entry visa to complete their trip.
As a tourist, most nationalities can get a maximum of two entries to Russia within one 30-day period per visa. Russia will also only permit one active visa per passport at any one time (so you can't enter three times by using a double-entry Russian visa AND a single entry Russian visa in your passport). Russia will also only issue visas from your country of domicile. These factors combined render any plans to visit Russia three times en-route impossible on standard tourist visas.
Everyone can apply for an auto-tourism visa for single or double entry. If you want to dip in and out of Russia more than twice, things get slightly more complicated. You could apply for a double entry visa and then apply for a transit visa in Ulaan-Bataar for your third entry. However, in doing this you are taking a risk and you're not guaranteed to be granted the transit visa.
The Visa Machine go into more depth in their help pages, here.