Guys, guys...
There is SO MUCH info in this thread that even I got confused for a minute...
Please allow me to clarify this, as an Israeli citizen who's also working in the aviation industry and is very familiar with the subject:
Out of the 245 countries/territories/independent states in the world today, only the following countries will probably not allow admittance to someone whose passport is stamped with an Israeli stamp:
Iran, Lebanon, Libya, Kuwait, Saudi Arabia, Syria, Sudan and Yemen.
Those are countries to which Israel is considered an enemy (and v.v), and although the passport holder is only a "third party", they will not approve of a previous visit to Israel.
THAT'S IT.
I should state though that other arab countries, such as Algeria, may cause some difficulties but it is not something official.
Apart from those,
nowhere else in the world would be a problem to enter when having an Israeli stamp in your passport. this also includes all other arab countries not mentioned here (yes, also UAE).
In fact - other arab countries (which although are not having an "open" peace agreement with Israel but are having diplomatic contacts) even allow nowadays Israeli citizens to enter their countries, including Qatar, Bahrain, Morroco, and even Iraq - and obviously those with peace agreements such as Egypt and Jordan.
And if you really do want to visit Jerusalem still be able to go someday to Riyadh as well, for example, many countries (such as Canada, the UK, and Singapore) do issue very easily a second passport for their citizens, which states "valid for Israel only" (since the problem would be the israeli stamp when you want to visit Saudi Arabia, and not the other way around).
These passports are not connected to your main passport in any way, and no one can ever know that you have one at home.
...But after all this, if you do find yourself standing in front of an immigration officer at Tel Aviv's airport, just ask them
not to stamp your passport.
Yeah, it's that simple.
The Israeli ministry of interior is aware of the issue, and therefore all foreigners that wish to enter Israel, can simply ask that their passport won't be stamped (and you can do this on entry as well as on departure). they will stamp a piece of paper says "entry permit", which won't even be attached to your passport, so you can simply discard it once leaving israel and no one should ever know you've been here
Apart from immigration, no other authority will ever stamp your passport, not even customs.
I hope that will end all the mixed info and help make things simpler...