Due to editing /system/build.prop and /efs/carrier/HiddenMenu you MUST be rooted. Also, you must be on a stock TouchWiz-based ROM. This will not work on CM builds/etc. At a very minimum, hiddenmenu.apk must be present on your stock device for this to work, but possibly others, so beware of ROMs that remove 'bloat' APKs.
1) Add the following line* to /system/build.prop
2) Change the contents of /efs/carrier/HiddenMenu from OFF to ON
4) Try it out! Go to the phone dialer and enter *#06# (show MEID) as a test.
For more codes that worked on the S3 that (may) also work on the S4, see below:
Having the hidden menus enabled is particularly useful for those who want to disable WiFi Power Saving mode -- I was having issues with an access point at work, but sadly disabling it didn't fix the problem. Otherwise, the hidden menus are useful to test your sensors/display/etc or just looking around in general. Enjoy!
For those that are NOT rooted, there is an alternative to get to SOME of these menus, but it is not a free app (currently $1.99). It's called "Sammy S Mode". Note that I am not affiliated with the developer, I just bought it to test if the menus were present before I dug deeper into finding out how to fully enable the hidden menus on my Verizon Galaxy S4. When I have time in the next day or so, I'll try to figure out how the app calls the menus, and update the post if I find a free solution for those that are not rooted.
* Note: I added the "sys.hiddenmenu.enable=1" line right after the "persist.cne.feature=0" one, but I don't think it matters where you add it. Just be careful editing this file. I managed to get my phone boot-looping, probably because I used notepad to edit the file and it added Windows CR/LFs to it, and had to use the NoWipe Factory Image posted here to restore my phone. The second time around I used Root Explorer's built in text editor and it worked fine.
In case anyone is curious how I figured this out, click here.
Mods: Thanks for moving this to the development section since it fits better here.
I wanted to compile a list of the ones that seem useful or new in general and working for me:
*#197328640# Service Mode
*#0*# Screen/Sensor/Vibration/Touch/Speaker/LED/etc testing -- even gives out raw values for light sensor, barometer, etc in real time
*#2263 RF Band Selection
*#66336 MDM Randump Mode (defaults to OFF)