Obiousely the main chance is that Beohner is bluffing and that he will bring a clean debt ceiling inrease bill to the floor of the House and it will pass mostly with the votes of Democrats. He would then probably lose the gavel to some new Speaker which will make us cry thinking of Boehner. The problem with hoping for this, other than hope not being a plan, is that Boehner won't cave till the last minute (he has made that clear) and may not correctly judge exactly when the last minute arrives. I will assume that the Republicans won't just pass a clean debt limit increase -- that they will demand at least a fig leaf.
Now if it is something which amounts to nothing like the last time when it was that the Sente doesn't get paid till it passes a budget resolution, this might do the trick. Obama might sign it.
I will assume that the rules are that Boehner muust have a case that Obama caved and Obama must have a case that he signed a clean debt limit increase.
If so, I think it would be a very bad idea to pass a clean continuing resolution. This would make it very hard for any Republicans to vote for a clean debt limit increase and survive. I think the third best hoe is bill which combines continuing resolution, a debt limit increase an da fig leaf. Let's say the fig leaf is elimination of theACA tax on medical devices (no huge deal).
The Democrats in the Senate can declare victory -- the Senate buget resolution already calls for elimination of the divice tax. Obama can say that he would have signed the bill if it consisted only of a continuing resolution (CR) and the tax repeal, since he doens't care all that much about the tax (and since he hasn't been as explicit about no bargaining on the continuing resolution). Boehner will say that Obama signed an unclean debt limit after all so he won. No one will believe him, but the Republicans won't eel so disrespected that they have to inflict the gavel on someone else.
Maybe this is not acceptable to Obama (Kevin Drum thinks it isn't and shouldn't be) . So I add the rule that the debt ceiling bill must be a stand alone clean debt ceiling bill with no other provisions at all. Then the approach would be as follows. The House passes a CR with a figleaf. The House starts work on a clean debt limit increase. The Senate passes the CR with a figleaf. It goes to Obama's deks and sits there -- he doens't have to sign or veto it immediately. Boehner says they have to pass the debt limit incresae ot get Obama to sign the device tax repeal which is the best thing evarrr. Obama refuses to say why he is still thinking about whether to veto and especially whether it has anything to do with the debt limit.
The debt limit passes the House. Obama signs the CR with the dirty figleaf (and beleive me I have recently seen filthy fig leaves). Then the Senate passes the clean debt limit increase and Obama as he promised signs a clean debt limit increase.
Maybe this is not acceptable tot he Republicans so let's go Rube Goldberg. The figleaf is attached both the the CR and the debt limit increase. The CR gets to Obma's desk first. When the debt limit increase get's to Obam's desk he notes that it just raises the debt limit and eliminates a tax which he eliminated the day before yesterday. So one provision amounts to nothing at all an he consisders it a clean debt limit increase.
OK the device tax figleaf is unacceptble to someone with a veto. New figleaf is a super committee. The report of the committee can't be mended of filibustered. This is what the Republicans got in 2011 in addition to legislated cuts and sequestration. They declare complete victory and reporters manage not to laugh. TheSenate passes and Obama signs the bill. The super committee fails to issue a report (of course) and no harm is done.
OK maybe this outcome is not acceptable to Obama as the principal that caucuses get nothing nothing at all for threatening the full faith and credit of the US Treasury. So the next day he gives a speach in which he notes that the report of teh committee can't be filibusterer or amended but it will be vetoed if it contains any provision he doesn't support on its own merits. Any provision at all. In particular if a dime is cut form Medicaid, Social Security or Medicare or if a dime is added to the taxes of any couple whcih makes under $250,000/yr or any individual who makes over $2000,000/yr he will veto the bill. To the committee he says "my way or the highway. And rather than waste your time why don't you take the highway and disolve. Then I will be willing to negotiate just as if the whole sorry debt limit impasse never occurred. I have nothing against super committees and I would even be willing to negotiate possibly setting up a new one untainted by debt limit extortion, but I will not compromise at all with a committee set up using debt limit extortion."
So the Republicans end up with exactly nothing. Obama wins. Boehner doesn't read this bog so he didn't see it coming.