From what I can tell, a lot more war trophies/ souvenirs were brought home by troops in WW2 compared to the first Gulf War. According to a chapter in the book pictured below, it was expressly forbidden by CENTCOM for US troops to take home any souvenirs, even a jar of sand, from Iraq. This really ticked off a lot of people as there was a well known picture of General Schwarzkopf holding up such a jar:
Image taken from the book "Crusade" by Rick Atkinson.
According to the book above, some troops tried to dispose of their forbidden souvenirs in dumpsters around camp. Items found included AK-47s, ammo and grenades. Yikes!
Anyway, here are some of my own souvenirs from Iraq (circa 2006):
Australian military issue Boonie hat in desert camo pattern.
A couple of patches. The left hand one was made by a local tailor. These were quite popular amongst the troops as the official issued flag patch was in full colour and hence a bit more conspicuous when worn with desert cams. The US rank patch was one that I wore on my body armour alongside my Australian rank patch, a practice that was commonplace as the US troops we worked with were often unfamiliar with our insignia.
Australian flag patch. Official military issue.
An old copy of Stars and Stripes.
Playing cards purchased from a PX in Baghdad.
Edit: photo of flag patch added.