This was me too!
After a quick Google search, I feel pretty stupid that I didn't know the word.
This seems mostly like a great question, but the constraint:
seems to ruin this question. When the bull gear is really large, you don't need another gear pass, which if we recall, each gear pass consumes 1-2% (or more) of your power, decreasing your efficiency. So I can make any ratio I want--even 1000:1 with only one gear pass, but the size of the transmission would be silly. So you can't disregard that stuff.
Also, each gear pass consumes dollars. Fewer passes are less money.
I remember the year we used andymark's "cimplebox"
http://www.andymark.com/CIMplebox-p/am-0734.htm which only has one gear pass in it. I was hoping that the gain in efficiency would help to compensate for a really bad gear ratio (4.67 to 1 max). We ended up with an amazing top speed but it was a terrible robot to position for frisbees.
Also don't forget that most of us that are designing transmissions are still using off-the-shelf gears from robotics vendors, but if we were custom designing gears, we could make them look like the beauties in the Chairman's trophy,
http://www.chiefdelphi.com/media/img...3509e9d6_l.jpg
with spans and webs and lots of weight reduction.
But for the complexity of obtaining your own gears, the weight reduction just isn't worth it.
Also, a 'real' gear designer will vary the modulus of elasticity (materials) of the gears in the train so point contact doesn't increase wear. Never see robots do this. We use steel gears.
So though I'd like to see some kind of comparison, but with no changing material weight and no lightening holes in the gears, it would be hard to be fair in the comparison. I think multiple passes would win every time because the bull gears would be so heavy.
A funny aside...once a supervisor asked me to put some 'lightening' holes in a part and I drew a blank. I responded "how big of hole does lightning make?"