What happens to mail weighing over one pound mailed in an outside mail box, with no return address? I know if such a parcel had a return address, it would be returned to sender, since you must take all U.S. mail weighing over a pound to the post office now. BTW, I am talking about harmless mail, not a terrorist plot or anything like that.
Thank you in advance to all who reply 
It would be sent to the USPS’ Mail Recovery Centers where someone would open the package in search of anything which might identify the sender or recipient. If this is unsuccessful, it is either held onto for some time pending a request from the sender and then destroyed. See an article like this for more: http://www.associatedcontent.com/article/25176/what_happens_to_dead_letters_at_the.html
I have sent mailers that are very clearly documents that have been over a lb with just postage. I clearly write the company to company addresses and for the most part have put them in the mailbox that is in the parking lot of the sending company. I have also sent them from the drop box inside a post office (where for reasons I won’t get into, they know me). I’ve never had a problem with late or delayed mail in any way.
I’ve also dropped them at other post offices. I’ve not had one returned yet.
Again, obviously all documents. All very well labeled. And the most important thing–traceable.
YMMV.
What, even expensive things like jewelry?
Australia Post (and also other outfits like the railway lost property office) holds periodic auctions of unclaimed valuables. Perhaps the USPS does likewise.
They, uh, deliver it?
Maybe you meant undeliverable mail, in which case iano seems to have the definitive answer.
Sorry, I overlooked this key phrase. :o Since I-don’t-know-when, any package weighing more than 1 pound is supposed to be mailed at the counter. You are not supposed to drop it in a mailbox. I can’t imagine why because they don’t follow any special procedures when you do take it to the counter. So it looks like BrassyPhrase has the definitive answer, which is that the package is delivered.
Fortunately, if you print your postage online, you can schedule a pickup at your residence for it or just drop it in the drop-off box (separate from the regular mail slot) at the post office as well, at least here in Austin. No need to wait in line at the counter.