The evolution of this odorless litterbox began with migrating from conventional litter to a sift-through box, using inexpensive wood pellets (either pine horse bedding or hardwood stove pellets, either one costing under $6 for 40 pounds). The pellets disintegrate when peed on, and the litter scoop is used to rake the resulting debris (wet sawdust) to drive it through the perforations of the nested box, into the lower one.

The problem was that the smell would still build, no matter how quickly I'd attend to new pee. And poop smells worst when it's freshest, so failure to get it out immediately resulted in stink.

This build is stage 2 of 3. I upgraded the bottom box to a deeper one to accommodate an exhaust port and to allow accumulation of more than a month of debris. Stage 3 will route the water output from my furnace/AC to the bottom bin, with a screen-filtered output running to a floor drain. This will effectively eliminate all accumulation of pee. [2023 edit: Nope. Needless. Perfect as is. Read on.]

Supplies

Anyone doing a similar project would depend on availability of items that would doubtless differ substantially from the things I had lying about, or was able to find at a surplus store. I'll just list the needed items.

A shallow and a deep tote of the same product line from the same manufacturer, assuring a good nested fit.
About a thousand quarter inch holes drilled into the bottom of the shallow tote, to allow debris through. A serious project could just buy some stainless steel perf and weld it to stainless steel side panels.
A hole bored into a side of the deeper, bottom tote to allow exhaust. Must clear the bottom of the shallow tote, must be high enough to allow deep accumulation of debris without blocking it (longer duration between emptying), and must not be vulnerable to wetting or wicking of pee that may fall upon the inserted hose. Many ways of doing this. Plan for how you want the box oriented near the wall, and how you'll need to route the hose to the window.
An old vacuum hose or something similar.
A chunk of insulating foam that'll fit your window and allow room for attaching the hose and mounting a fan.
A muffin fan. I went with a 12VDC unit that consumes little power, and moves relatively little air. It only needs to ensure that no air near the surface of the pellets escapes the box. Tip: level the pellets well with the scoop when tending the box, to ensure uniform drawdown.
A power supply for the muffin fan. These latter two items are really cheap at any good surplus place. Consider where a power jack is in whatever wall you'll be doing this on. Your cord will need to reach the window.
Options as necessary. I used a yogurt lid to keep the rain off the fan, but I'll likely be changing how that works at some point.

Step 1: Nested Boxes

If you've never done nested boxes with pellets, it's pretty cool -- but doesn't get rid of the smell. The ventilated approach gets rid of ALL the smell. So get a couple boxes -- one deep, one more shallow (but not so shallow your male cat will pee over the top; note how I used the lid of the tote to protect the wall and guide any pee spray into the box).

With the setup I have, I can go a couple months without removing the debris from the bottom tote. Interestingly, the constant ventilation evaporates much of the pee (the sawdust in the bottom increases the wetted surface area dramatically, speeding the evap). I end up dumping this in my compost pile (no poop allowed, results not used near food crops).

You'll need to drill a lot of holes in the bottom of the nested tote. Be systematic, take your time. Too far apart and each tending will take longer (to scrape it about and drive the pee-debris through to the bottom). Too close and you'll weaken the bottom. Drill carelessly and you may create cracks; with that in mind you'll want a tote not made of brittle plastic; something softer. Quarter inch holes are good, and a step bit seems pretty friendly for getting the job done fast without the bit tip drifting across the surface.

Step 2: Running the Hose

Having a "socket" to plug the hose into might be better than just having a round hole. I found a pill container was a perfect fit for the cheap hose I used. Carefully cut a hole in the tote to fit that, then use plastic JB Weld inside and out (after buffing the tote surface with a bit of sandpaper) to mount the socket. You might also consider mounting the hole at a different angle; I just chose default perpendicular to the side of the tote.

The other end of my hose fit a paint can top, so I cut off the top of the top and mounted it in a proper-sized hole I cut in the foam. You could just as well stick it directly into the foam, but I was concerned about good integrity for my fan mount.

Step 3: Muffin Fan

I mounted the muffin fan with four small machine threaded screws, with nuts on the outside and extra screw length allowing me to mount a plastic rain-guard while allowing enough room for air movement. You can't really tighten the mount to the foam, and you don't need to (use big washers if you must).

Finally, the older of the two cats didn't like the way my deeper tote had raised the litter box so much, so I used the old lower tote as a step. Did the trick.

That's it. After the first day at work, I came home and smelled nothing at all. It was amazing. I could get my face within 8 inches of the pellets and not smell a thing. No air wafts upward from the surface of the litter at all. It's just a gentle draw-down that's sufficient to prevent any odors from escaping into the house -- and not so much flow that it'll materially increase the heating/cooling bill.

It's been over a month now, and I think I can wait another couple months before cleaning out the bottom tote.

As indicated, I may consider "flushing" the debris by routing the condensate from the furnace/AC to the box, with a screen-filtered output running into a floor drain that's very near the box. It wouldn't smell, because the volume of water is substantial. The box would probably get a dozen full flushings (about a quart each time) between each occasion of the cats peeing. However, the effectiveness would depend on strategic distribution of the flushing water so that all the accumulating sawdust gets well-wetted with the flush water. This challenge leaves me unmotivated to move on to that configuration anytime soon. There's an "if it ain't broke, don't fix it" aspect to this thing. Just now, it seems nearly perfect.

Best of luck with your own variation on this.

Step 4: 2022 Update

I'm going to skip the flushing. Adds needless complexity to something that's working perfectly. Only hitch so far was when I added too much pellets, which prevented good draw-through of air. This resulted in some odors wafting off the bed of litter again.

Pellet litter normally has one draw-back; namely, even though pee generally turns it to sawdust so that it may be sifted into a lower tote, some pee still contaminates pellets that don't get enough on them to fall apart, but they'll end up stinking. Over time, the entire bed of pellets can get to smelling darned nasty.

This system makes that entirely moot, I've discovered. It doesn't matter if the pellets stink to high heaven -- come to think of it, that's the only place they'll stink to since that smell is driven outside. ;-) It occurs to me that this raises the stakes for power outages, though. Might want to put a backup power supply on the fan power. ;-)

Another effect of not worrying about smelly pellets is that I'm consuming so darned little of the stuff. When I converted to pellets I realized I was saving a lot of money on litter. But that presumed tossing a boxful of pellets when they got smelly. Since that's not a problem anymore, I never empty pellets from the box. The result is that a 40# bag of pellets ($6 at most) will likely last me 6 months or more. It's ridiculous how much this saves. This certainly offsets whatever increase in natural gas consumption I pay for on account of exhausting house air at a very small rate (which means cold air is drawn in and needs to be heated, or in summer the warm incoming air needs to be cooled).

I keep a five gallon bucket outside the back door (with a small trash bag and loose-fit lid) for scooped poop.