Additionally, if let's say you are executing a search warrant for dope and find a dead body. You really expect the courts to say 'no, pretend you didn't find it?' The smart officers stop right there (where they found the body) and go get another warrant. Just as the guy finding the illegal magazine might want to stop and try to get a warrant for things related to manufacture or possession of illegal weapons.
The danger in searching further is tainting related evidence found after the discovery of the body or the magazine.
Exceptions to the search warrant: search incident to arrest - limited to area in immediate control of the subject arrested; you can enter and search without a warrant based on exigent circumstances - hostage rescue, 911 DV hang up, etc.; and you can most likely search a vehicle w/o a warrant based on probable cause under the Carroll Doctrine; another exemption is upon consent from the person who has legal standing to consent to the search.
The 4th is probably the most litigated Amendment, while it is hard to order then, I personally believe the 4th is our most precious Amendment. Followed by 2nd and 1st. Personally.