Powershell, <s>73</s> 55 bytes

Huge thanks to TimmyD for shaving off 18 bytes!




 `$n` is **n** in **x<sub>n-1</sub>**

Explanation and exploded code:

    $A=1111                            #starting number
    $n=4                               #n in formula
    for($i=0; $i -lt $n;$i++)          #loop n times
        [string]$B=[math]::pow($A,2)   #create a new string $B and set it to $A raised to the power of 2
        $A=$B.substring(0,4)           #set $A to the first 4 characters of $B
    $A                             #print $A

Some notes:

 - Powershell lets you assign variables in the same statements where you reference them. For example, `1..($n=4)|%` will set $n to 4 and then start a loop that runs $n times. `1` can be changed to any integer and it will loop $n-[your integer]+1 times.
 - The default data type when using `[math]::` in Powershell is a double. In the code above, we have to explicitly cast `$B` to a string so that we can call `.substring()` on it because there is no `.substring()` function for doubles in Powershell.