What's a good ratio for clothes to days on a long backpacking trip, or to Philmont? For instance, how many clothes should I bring for two weeks?
At least one of everything, an extra pair of undies, and as many socks as you want.
You're going out into the backcountry, not staying at a Hilton. If you're going to be out on a long trip, bring a bag and some soap to wash your clothes if they get too soiled. The last thing you want on a long trip is more weight on your back. Bringing multiple pairs of anything (except undies and socks) is not necessary. The only time I pack more than one outfit is during the winter, when the necessity to get out of wet clothes and into something dry could mean the difference between life and death.
So at least one of everything, and at most two of everything.
Personally, I keep it simple. One long sleeve shirt for daytime use, one long sleeve shirt at night. I wear one set of boxers in a week, and a pair of long underwear at night (if it's under 50f).
One pair of pants I can roll up. Three pairs of socks per week. One to wear, one for bed, and one that's been rinsed out and drying on my pack.
Bringing a beanie is usually a good idea. For rain, that's a whole topic.
Most new people bring way too many clothes. Wool stinks a lot less than synthetics, so if smelling like ass is a concern, go with a lot of wool. Particularly your shirt.