My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.

       
#include <iostream>
       
 
       
// return (base^exponent) % modulo for 32-bit values, no need for mulmod

       
unsigned int powmod(unsigned int base, unsigned int exponent, unsigned int modulo)
       
{
       
  unsigned int result = 1;
       
  while (exponent > 0)
       
  {
       
    // fast exponentation:
       
    // odd exponent ? a^b = a*a^(b-1)
       
    if (exponent & 1)
       
      result = (result * (unsigned long long)base) % modulo;
       
 
       
    // even exponent ? a^b = (a*a)^(b/2)
       
    base = (base * (unsigned long long)base) % modulo;
       
    exponent >>= 1;
       
  }
       
  return result;
       
}
       
 
       
// compute result

       
unsigned int tetration(unsigned int a, unsigned int b, unsigned int modulo)
       
{
       
  unsigned int last   = 0;
       
  unsigned int result = 1;
       
  while (b--)
       
  {
       
    result = powmod(a, result, modulo);
       
 
       
    // converges pretty fast, abort early
       
    if (last == result)
       
      break;
       
    last = result;
       
  }
       
  return result;
       
}
       
 
       
int main()
       
{
       
  unsigned int a = 1777;
       
  unsigned int b = 1855;
       
  std::cin >> a >> b;
       
 
       
  unsigned int modulo = 100000000;
       
 
       
  // abort if gcd(a, modulo) != 1
       
  if (a % 10 == 0)
       
    return 1;
       
 
       
  std::cout << tetration(a, b, modulo) << std::endl;
       
  return 0;
       
}

This solution contains 8 empty lines, 7 comments and 1 preprocessor command.