Avoiding Israeli Stamp
There's lots of confusion about this, and little information (or I've been a bad "Hubber" and simply haven't searched hard enough). Here's what I've just done (with much thanks to Gareth who did it a few weeks before me!)...
Crossing from Jordan to Israel via the King Hussein Bridge, it IS possible to avoid an exit stamp from Jordan and the Israeli stamp in your passport. (This works because Jordan considers this territory Palestine, not Israel, or something like that. In fact, they may even consider it as part of their territory. I'm not sure.) Anyway, it is NOT possible to ride across this bridge.
So, in order to avoid the stamp, you must leave your vehicle in Jordan at a pay-park place (they're easy to find) and cross without your bike. BOOO!
But it's still not simple. You will likely be told by Israeli border guards (as I was) that EVERYONE gets a stamp in their passport and that there are NO exceptions. But, if you find the right person in the right mood and have a convincing story (like, you're continuing to Africa with a motorcycle and you need to pass through Sudan or Libya) you can cross with no stamp.
Of course, that leaves you in Israel without a motorcycle. And you must leave Israel via the same border crossing before your Jordanian visa expires to avoid having Jordan stamp you back in or issue a fresh visa (which would be a dead giveaway that you've been to some "mysterious" place.)
I hope this helps. And, I hope it works for me... I'm still in Israel (with no stamp!) and haven't made it back to my bike, yet. We'll see.
Has anyone (within the last, say, year) made it into Israel WITH their bike and WITHOUT a stamp? I'd love to hear about it!