Calculate Knuth's up arrow notation

Java 7, 121 bytes

int c(int n,int a,int m){return(int)(a<2?Math.pow(n,m):d(n,a));}double d(int n,int a){return Math.pow(n,a<2?n:d(n,a-1));}

Explanation:

int c(int n, int a, int m){         // Main method with the three integer-parameters as specified by OP's challenge
  return (int)                      // Math.pow returns a double, so we cast it to an integer
          (a < 2 ?                  // if (a == 1):
                   Math.pow(n, m)   //  Use n^m
                 :                  // else (a > 1):
                   d(n, a));        //  Use method d(n, a)
}

double d(int n, int a){             // Method d with two integer-parameters
  return Math.pow(n, a < 2          // n ^ X where
                      ? n           //  X = n    if (a == 1)
                      : d(n, a-1)); //  X = recursive call d(n, a-1)    if (a > 1)
}

// In pseudo-code:
c(n, a, m){
  if a == 1: return n^m
  if a > 1:  return d(n, a);
}
d(n, a){
  if a == 1: return n^n
  if a > 1:  return d(n, a-1);
}

Test code:

Try it here.

class M{
  static int c(int n,int a,int m){return(int)(a<2?Math.pow(n,m):d(n,a));}
  static double d(int n,int a){return Math.pow(n,a<2?n:d(n,a-1));}

  public static void main(String[] a){
    System.out.println(c(2, 1, 2));
    System.out.println(c(2, 1, 3));
    System.out.println(c(2, 2, 3));
    System.out.println(c(2, 3, 3));
  }
}

Output:

4
8
16
65536

JavaScript ES7, 53 44 bytes

f=(a,b,c)=>b<2||c<1?a**c:f(a,b-1,f(a,b,c-1))

f=(a,b,c)=>b<2||c<1?a**c:f(a,b-1,f(a,b,c-1))

console.log(f(2,1,2));
console.log(f(2,1,3));
console.log(f(2,2,3));
console.log(f(2,3,3));

Mathematica, 48 40 bytes

If[#3>1<#,#0[#0[#-1,##2],#2,#3-1],#2^#]&

Order of arguments is m, n, a (using the notation from the challenge).