It's just possible that President Bush could remain in office after the election with John Edwards as his vice-president. I just did some calculations, and if all goes according to the following outline, that will be the result.
If the electoral college ties (or if no one reaches 270 electoral votes due to a state's vote ending up with an exact tie, which is extremely unlikely but possible), then the election goes to the Congress. Perhaps the most likely way for Bush and Kerry both to end up with 269 votes each would be if Bush were to win everything he had in 2000 except Nevada and New Hampshire and Kerry to win the rest. In that event, the House of Representatives has the responsibility of choosing the president and the Senate the vice-president. It's not a strict vote, however, so it doesn't mean whoever controls each will elect its candidate. Each state gets one vote in each election. I looked through how many people of each party occupy the House seats from each state, and looked at some of the closer House elections to see how those might affect the vote, and it seems the Republicans clearly have a major advantage there. My estimation has Bush winning 28-30 votes and Kerry winning 15-18. A few states would or might tie, depending on who wins the elections this time around. So Bush really only needs 269 votes to win the presidency. Kerry needs 270.
The vice-presidential runoff in the Senate is what's really interesting. It could go either way, depending on who wins the close Senate elections. More than a few of these elections are way too close to call. If the Republicans win all the really close Senate elections, they'll control the votes for 22 states, with Democrats controlling 16 (also assuming a Senator Chafee vote for Bush, which might he might be talked into given the right pressure, though he is a Republican the way Zell Miller is a Democrat). The other states would tie. If Republicans lose all the really close elections, the Democrats would have 18 and Republicans 17 (assuming Senator Chafee goes with Kerry, which I think is more likely). That means the Democrats might be able to pull off a vice-presidency for Edwards if everything goes their way in the Senate elections and Kerry manages to get exactly 269 electoral votes. I know this isn't incredibly likely, but it's well within the realm of possibility. What that vice-presidency would look like would be a little odd. He'd be the tie-breaking vote in the Senate, but this would already be a Democratically-controlled Senate. He probably wouldn't have much role beyond that, though I'd imagine Bush would have him at all the cabinet meetings as a dissenting opinion, if only because it would look really bad if he didn't. You could get more non-partisan cooperation in the Congress with this sort of situation, but that's hard to predict. All of the above (except for Chafee, whom I mentioned) assumes people will vote with their party. I don't know of any other specific examples of people in Congress who will remain in Congress who support the candidate of the other party, so I didn't factor that in. If anyone knows of any, I would welcome that information and adjust my conclusions if necessary.
Of course, if the Senate vote to elect the vice-president ties, which is slightly more likely, then the House-elected president, who would be Bush, would just appoint Dick Cheney to the vice-presidency, and if the Senate vote is high enough then both Bush and Cheney would be elected by the Congress. In all of these cases, I'm sure some vocal idiots would cry foul, saying the election was spoiled, the president and/or vice-president would be illegitimate, unelected etc. This is the Constitution's set directive for such situations, however, and someone elected by the Congress in such a situation is an elected official according to all legal rules. That kind of election is closer to what the founders had in mind anyway, since they rightly didn't trust uninformed citizens to vote in a fully democratic manner.