A:A compromise between a really stiff bike and a really light bike would be a sort of stiff, sort of light bike, no?
But, you ask a serious question. A lot depends on the bike's geometry and other factors, but as a general rule aluminum is about the stiffest material commonly used in bicycle frames. Aluminum bikes are famously jittery; rides that I take on my aluminum bike feel completely different on my steel Eddy Merckx. Like a different road. Of course, stiffness also is aluminum's great virtue. It allows frame makers to use extremely thin-walled tubing, cutting weight, although that is somewhat offset by the need to prevent flex, hence thick-diameter tubes. So aluminum bikes are typically very light, yet because aluminum is not terribly exotic, they're not real expensive compared to bikes made from such materials as carbon fiber or titanium. Aluminum frames' stiffness also helps make them good climbing and sprinting bikes - you crank on the pedal, and just about all that energy goes from the pedal to the wheel, not in bending the frame sideways as happens on softer bikes.
That said, the best compromise between stiffness and lightweight would appear to be scandium, an aluminum alloy developed by the Russians for military aircraft and now found in some bikes made by Fuji, Merckx and a few other makers. It's lighter than titanium on a same-volume basis, and has flex characteristics more like steel than aluminum. And it's not wildly expensive - about $1,500 to $2,000 for a frame. That's not cheap, of course, but when a Litespeed Vortex goes for almost $3,000, it begins to look pretty realistic.
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