Not so many countries that we traveled to in the Middle East (not our fault--most of them didn't want us), so not many obsessions. Here's what we have, though. As a reminder, we included Egypt in the last Africa obsessions post, though it is arguably more like the Middle East than like Africa.
Jordan- Sweets. Most of the Middle East has a sweet tooth, but it is most pronounced in Jordan. In addition to tons of regular candy shops selling everything from candy coated almonds to gummy rings, most of the sweets come in two basic baklava-like types: fresh and dry. Fresh are generally not so good, but the slightly drier ones (shelf stable, nonetheless), are really, really good.
Israel- Security. Probably an obvious one, but after practically strip searching me on the way into the country (before they had even decided if I could come in), and having to have our bag x-rayed to go into a supermarket, this is definitely the only choice for Israel's obsession. Well, unless being really expensive is an obsession...
Turkey- Sensor Lights. Is Turkey part of the Middle East? I say yes until they are part of the EU. What's a sensor light, you ask? One of those lights that only comes on when there is movement, like some people use for security lights. Most of Turkey's lights seem to be sensor lights, especially in hallways. Most annoying, though, are the bathroom sensor lights that turn off if you stand at the urinal for more than about 10 seconds. Sure, you just have to move to turn them back on, but it's hard to wave at the lights if your hands are otherwise occupied.
Short and sweet. Hope that you enjoyed!