You've asked two entirely different questions. Each has an entirely different answer.
What's the most you can bid down on a house? As much as you want. I've know investors who've bought houses for 60 cents on the dollar. A house is worth $400,000; they've bought for $240,000. Or less. And those aren't short sales or foreclosures. They're motivated sellers who want to sell quickly for cash. There aren't a lot of those deals out there, but there are some. And the point is: There's no "floor"--no reverse cap of 10%, or 20%, or whatever.
Your second question is "What's the lowest reasonable amount?" Well, what's "reasonable"? If you told me: "A reasonable offer is one that the seller accepts or counters 90% of the time" (I just made that definition up, by the way), then depending on the market you're in, it might be anywhere from 2%-6% off the list price. But I don't like list prices; they may be higher or lower than the actual value. What you should be aiming at is buying a house at or below what the comps suggest it's worth. So, determine the house's value via comps, then consider coming in somewhat under that.
Hope that helps.