Rooting the Galaxy S3 Without a Computer Is it possible to root the Galaxy S3 without a computer? If so, what would the process be? My Galaxy S3 is with Verizon and is currently running Android 4.3.
You can indeed. 

DISCLAIMER: Rooting may brick your device and invalidate your carrier and manufacturer warranty. I didn't write the guide below (full credit is due to the poster on XDA Developers, and I have left the words 'my' in place but it is HIS work), and it is your responsibility to check you are happy to proceed. HERE BE DRAGONS. 

This guide was written in July of 2013 and I can't say for sure if it will work on Android 4.3 and your particular bootloader at all. I have not tested this method. 

Rooting form a PC is easier and has less scope for human errors and is liklier to work in general. 

 This XDA thread is relating to the sprint version but should apply to the S3 general. 

Prerequisites

The following files are attached to this post for convenience. Download and install the apps, download and extract the zip file somewhere convenient.
Superuser by Koush(MUST be this one. You can change it later if you so desire.) 
Terminal Emulator by Jack Palevich ( MUST be this one) 
Zip file, extracted somewhere convenient 

THE GOODS

These generic names will be used:
/path/to/motochopperDir - This is the full path to the extracted motochopper-mobile DIRECTORY or FOLDER. Example: /sdcard/motochopper-mobile (will vary depending on where you extracted the files).

NOTE: After running the script, Motochopper may or may not print "Failed", but the exploit DOES indeed work regardless of this. Make sure you reboot for changes to take effect.  

Download the attached zip file AND both apps. 
Install Terminal Emulator and Superuser (use a file manager, navigate to the downloaded .apk files and click on each file to install them). 
Open up Terminal Emulator and type the following commands. Press the ENTER key after each command.

Code:

cd /path/to/motochopperDir
cat setup.sh &gt; /data/data/jackpal.androidterm/setup.sh
cd /data/data/jackpal.androidterm
chmod 755 setup.sh

./setup.sh /path/to/motochopperDir


From there, my setup.sh script will handle the rest of the process. If you are still unsure, see my sample run below.

Sample run (files were extracted to /sdcard/moto)

cd /sdcard/moto
cat setup.sh &gt; /data/data/jackpal.androidterm/setup.sh
cd /data/data/jackpal.androidterm
chmod 755 setup.sh
./setup.sh /sdcard/moto


The Links:

Android Term.apk

Super User

MotoChopper