I think the Beige G3 desktop is your best choice for the requirements you mentioned. It's RAM is limited to 768MB instead of 1024MB like most of the other choices, but it really is a faster machine. It's architecture is just more efficient and faster than the older x500/x600 machines. The Beige also won't run any OS earlier than 8.0. so if you need 7.6.1 or earlier, this isn't your machine. The 1GHz G3 ZIF is still available from OWC for a little over $100 the last time I checked.
If you don't need a machine that will run OSX, you should consider quite a few of the clones as well. Sticking to desktop machines, since you mentioned scooting it under the bed, and eliminating the machines with RAM limitations under 200MB, that would include the Umax J700 (four PCI slots, easily upgradable to six and 1040 (or 1056?) MB RAM capacity), Power Computing PowerCenter and PowerCenter Pro. However, the PowerTower Pro is a purely ATX based form factor, so you could get a PowerTower Pro and put it in a desktop style ATX case. Actually, the same is true of the J700's fraternal twin, the S900.
Bah, this is getting messy. If built-in video isn't mentioned there is none.
1) Desktop cased machines with upgradable CPUs:
A) which will run OSX (with xpostfacto):
1. Apple:
PM7500, PM7600, PM7300: 1024MB RAM, 3 PCI slots, built-in video (may not work on OSX)
2. Umax:
J700: 1040 (or 1056) MB RAM, 4 PCI slots, easy soldering to add two more PCI slots
S900: As above, but comes standard with 6 PCI slots. In a tower case, but easy to move to an ATX desktop. Actually, the stock case is convertible to desktop mode, but requires a new bezel which probably isn't available any more.
3. Power Computing:
Power Tower Pro: 1024MB RAM, 6 PCI slots, comes in a tower case but easy to move to desktop ATX case.
PowerWave: 1024MB RAM, 3 PCI slots, comes in a tower case and is challenging to move to ATX desktop case because PCI slots are horizontal on riser card.
B) which probably won't run OSX (unless xpostfacto added support since I last checked.)
1. Power Computing
PowerCurve: 512MB RAM, 3 PCI slots, built-in video
PowerCenter: As above
PowerCenterPro: As above but available in both tower and desktop cases.
2. A number of other clones, but they all have RAM limitations below 200MB.
2) Without upgradable CPU and probably won't run OSX
A) Apple
PM7200: 512MB RAM (really 1024MB, but no-one ever made 256MB, 5V, FPM, buffered DIMMs), 3 PCI slots, built-in video. Available in 75, 90 and 120 MHz models.
BTW, the 7500 and 7600 use slightly different power supply connectors, IIRC. So don't assume you can mix and match logic boards with power supplies, if you end up building from parts. I think the 7300 uses the same power supply connector as the 7600. The 7300 lacks video capture (no big loss in my opinion) and has the updated ROM which is found in the non-Kansas, original 8600 and 9600. And no, I've never been able to figure out what difference it makes. But if you can't figure it out any other way, you can look at the four ROM chips (back of logic board, four chips ~.5" X 1.1") and the original x500, 7600 ROM is labeled 341S0168 through 341S0171, whereas the 7300, 8600, 9600 ROM is labeled 341S0280 through 341S0283. The Kansas, 8600/9600 Enhanced ROM is labeled 341S0380 through 341S0383.
And I may have misremembered and all the 341s above maybe are really 343s....
Most of the clones use a bog standard ATX power supply, but the J700 and S900 don't. It's ATX, but the connectors are like the old AT power supply connectors with one 3 pin connector like fans use. Instructions for converting an ATX power supply for use on the Umax machines here:
http://www.kennedybrandt.com/supermac_insider/support/psconversion.html but the bit about the BMI connector working with the mini-fit Jr. is wrong. I need to correct that some day.