<h2>Javascript</h2>

Here's the basic format of all the solutions.  This case returns `true` if `a>b` and `false` otherwise:

    function greaterThan(a,b){
        c=a;//Copy a
        while(true){
            if(b==a){
                return false;
            }
            if(b==c){
                return true;
            }
            a++;//Approaches b if a<b
            c--;//Approaches b if a>b
        }
    }


**Warning:**  Since we didn't need to support the `A==B` case, the next 3 solutions will lock up for those inputs.  Thus, don't fill in equal or empty numbers at the prompts in the test fiddles, unless of course you like pushing red buttons that say "Do **NOT** push!!!"

<b>149 chars</b> - [Fiddle][1]: Conforms EXACTLY to specs ;)

`for(A=B='A',C='B',a=c=prompt(A),b=prompt(C);b!=c++||(B=C),b!=a--||(A=C),A==B;);alert("Integer "+B+" is Greater than "+A+".\n\n\nRules  : No Brainfuck.")`

<b>74 chars</b> - [Fiddle][2]: Prompts user for 2 numbers, alerts '-1' if 1st<2nd, '1' if 2nd>1st:

    for(A=0,a=c=prompt(),b=prompt();b!=c++||(A=-1),b!=a--||(A=1),!A;);alert(A)

<b>54 chars</b> - [Fiddle][3]: Function that takes 2 parameters and returns `true` if `A > B`, `false` if `A < B`, and, of course, locks up if `A==B`.

    function a(a,b){for(c=a;b!=++a&&b!=--c;);return b!=a;}


  [1]: http://jsfiddle.net/briguy37/aEDX6/
  [2]: http://jsfiddle.net/briguy37/DxpwK/
  [3]: http://jsfiddle.net/briguy37/SK3X2/