So I collect my materials over the course of days or weeks--months for the monster piles.

Junk mail comes in EVERY SINGLE DAY. You can stop a few things by trying to remove your name, but total elimination is not realistically possible.

Every piece of paper that I'm done with (all the junk mail, child's school papers, old user's manuals, etc.) gets twisted into a long paper twist and chucked into the compost. The twist helps to keep the paper from matting and provides air space between everything. I get the local newspaper, so that gets torn into strips and tossed in the compost. The shells from the chicken eggs are crushed and tossed in there. Coffee grounds and filter? Bonus! Used facial tissue, facial tissue boxes, cardboard toilet paper or paper towel rolls, food scraps, potato peelings, the frass from my mealworm operation, essentially ANYTHING that is not meat, not plastic/metal/inorganic and isn't contaminated with fecal matter of a meat-eating animal goes into the compost.

Our family of three plus chickens, horse, cat and dog rarely fill even a quarter of weekly trash can. (We do also have a recyclables container that gets far more use).

I'm considering hitting up the neighbors and maybe a local restaurant or two for their food waste (which gets mixed with the drier matter). Our food wastes are currently split up between at least 1,000 mealworms, 35 chickens, unknown numbers of redworms, and our many compost piles, and there just isn't enough to go around.

I don't start a new pile anymore, I only removed the finished and keep the pile going. Starting a new pile does take a lot of stuff, but if you do it once you can just keep on going, if you are not seasonal bound by the elements.

That's a great blanket statement, but could you qualify that with something? Do you have a source to validate the blanket statement? Do you discriminate between office paper and glossies?

I compost all the envelopes (sans the glassy window) and all the non-glossy paper in my junk mail. Glossy papers go into the recycling, stupid fake credit cards and fake keys go into the trash (I need to start calling to yell at those dealerships).

I have found that leaves + grass clippings = hot pile

Even the largest plastic bins you can buy are only about 1/2 to 2/3 of cubic yard, and they work just fine.

Even a small pile will turn into compost, it will just happen more slowly.

Among the most important things not to do is put anything in your wood stove that shouldn't be burned. It can not be stressed enough, so we'll say it again: except for a small amount of newspaper (no colored, slick inserts!) to start the fire, never burn anything other than seasoned firewood in a wood stove. Junk mail, bills, glossy magazine pages, and wet or rotting wood may seem like reasonably good fuel, but they all burn poorly and release high levels of particulate matter, which contains dangerous particles that can enter your lungs but can't escape. This can cause a host of health problems, including asthma and other respiratory diseases and cardiac issues and they can even damage your immune system.
Other items commonly burned in wood stoves are even worse. Never burn any of the following:
household trash
plastics
cardboard
Styrofoam
driftwood (especially from salt water)
pressure-treated or painted wood (including varnishes, sealants, and wood sprayed with pesticides)
plywood or particleboardIn New York State, as of October 2009, "burning household trash, whether in an open pit, burn barrel or a wood stove is illegal." That includes any of the items listed above. (For more information, go to the Department of Environmental Conservation's [DEC] web site.) There's a good reason it's illegal: the number of toxic chemicals these materials release into the air when burned could make your head spin-and literally could land you in the hospital. They include:
dioxins
spin-and literally could land you in the hospital. They include:
dioxins
volatile organic compounds (VOCs) (such as benzene, toluene and methyl chloroform)
furans
halogenated hydrocarbons
chlorinated fluorocarbons
carbon monoxide
carbon dioxide
sulfur dioxide
heavy metals (including lead, barium, chromium, cadmium, arsenic and mercury)

The average background compost pile should be a min of 3'x3' by 2' high, up to a max of 5'x5' by 2' high, IMHO. Your exterior carbon material can determine the size. With sawdust, the air travels less, so maybe 3'x'3'. With shredded leaves, 4'x4' should be good. With wood chips as the major ingredient, 5'x5', possible larger. I have two piles, each about 4'by8' by 2' high. A proper mix will give a core temp of 150F, at least in the warm months.

The often-repeated so-called optimium pile is FROM 3'by3'by3' TO 5'by5'by5' when the person wants the opportunity to make a hot pile. The person does not have to make a hot pile, but the bin should be large enough to retain heat if the person wants to. And those number probably came from someone who wanted nifty sounding numbers.

One last thing. While heat will speed breakdown, and kill some weed seeds, most piles can reach 160F+. But Cornell says there is faster breakdown at 130F than at 150-165F, where some of the good bacteria are being killed off.

driftwood
plywood
styrofoam
plastics (except for the little window)
painted surfaces or surfaces sprayed with insecticides

We actually use a wood-burning stove for primary heat. Wood-burning stoves actually take a bit of intelligence and skill to operate without smoking up or burning down your house. Most of those who use one know not to be total idiots and use it to burn dangerous crap.

Getting back to what actually IS in junk mail, which is predominantly a lot of simple white paper, and occasionally some glossies.

Glossies go to the recycle.
Paper goes to the compost.

"Most glossy paper, including magazines, catalogs, and colored newsprint, no longer contains heavy metals and can be added to your compost pile in small amounts. Because the paper is mostly cellulose, it will break down slowly and use up a good deal of nitrogen in the process. Shred the paper to speed decomposition and to keep it from forming an impenetrable layer in the pile."

Answer - you don't have to.

Composting is done in almost as many ways as there are composters. Relax - stuff rots whatever you do to it. You don't need to fill your bin all in one go. You can fill it as and when you have materials. It won't compost as fast but it will compost. If you have 2 bins you can fill one gradually then, when it's full, empty it into your 2nd bin and start filling number one again. A lot of posters here are compost perfectionists, some are not. I'm in the latter group. I fill the bin when I have stuff. I don't turn. I spread it when I feel like it. I am not bothered by heat or speed. Whatever works for you is the 'right' way to do it. For me composting is a practical way of dealing with kitchen and garden waste and getting a valuable dividend - it's not a life project.

And billme, You said, "I get almost no junk mail,"
Where do you live...Land's End?...hah!

Browns, or Carbon rich ingredients:

Is there a better ingredient than leaves? Not in my opinion. All of my neighbors collect their leaves bagged on the curb, and a very few even shred them first! I think of paper as the MacDonalds, and leaves as the fruit/vegetable/meat meal. Leaves keep long periods without problems.

Green, or Nitrogen rich stuff:

Your fruit/veggie wastes.
Those of your neighbor as well. Kids probably would take more of an interest than adults.
Grocery stores have been less keen over the last few decades to let dumpster divers paw over these goodies. Most store use closed systems. There are a few types of stores where one might be allowed to dumpster dive. Two years ago I quit packing my small hatchback with 600 pounds of green stuff, none of it yet smelly. I quit because my soil was improving, and it was too much work to cut it all up. So I will give you my best advice, seafood and Asian grocery stores, if you are looking for a LOT of produce.
And lastly, the coffee store. Starbucks is the famous one, but others may also give away the used grounds. Interestingly, I found the large bags (later small 10# bags) of Starbucks expresso grounds was not the type that worked out best. The larger bags with dryer grounds with filters, mixed with trash, was easier to mix with leaves and other materials.

You probably will have to have some type of relationship/friendship with one or more employees of a grocery or coffee store, to see that you get your share of the goodies. The Starbucks nearest me used to pitch 75-90% of the used grounds even when they were promoting the grounds-give-away. When I was going to the last grocery store giving away greens(I could get 1000# just of bok choy), I would have viewed any other dumpster diver looking at 'my' dumpster, and considered composting him/her.