Well, in our parish, it's held without fail in Great Lent and it's called "Soborovanie". Yes, I know there are technically supposed to be 7 priests in attendance, but, even though our parish is huge, it's very rare that we manage to have more than 3 priests at the service. Yes, the service is extremely lengthy, and there are usually such huge crowds of people that it takes literally hours for everyone to be anointed. Why then is it always held during the later half of Lent?
I am assuming then that it IS acceptable for a person to request a private anointing service?