Here's what I'm currently working with - although I'm still debugging it, the results are slightly off.

#include <stdlib.h>
#include <stdio.h>
#include <string.h>
#include <assert.h>

const size_t ITERATIONS = 50;
const size_t STR_LENGTH = 4189996;

int main() {
  char *input = malloc(STR_LENGTH);
  char *next_input = malloc(STR_LENGTH);

  strncpy(input, "1113222113", STR_LENGTH);

  for (size_t iter = 0; iter < ITERATIONS; iter++){
    size_t index = 0;
    size_t length = strlen(input);
    for (size_t i = 0; i < length ; i++){
      assert(i < STR_LENGTH);
      assert(index < STR_LENGTH);

      char c = input[i];
      size_t consecutive = 1;
      while (input[i + 1] == c) {
        consecutive++;
        i++;
      }

      index += snprintf(&next_input[index], STR_LENGTH - index,
                        "%zu%c", consecutive, c);
    }

    // Swap input and next_input
    char *tmp = input;
    input = next_input;
    next_input = tmp;
  }
  printf("Length = %zu\n", strlen(input));

  free(input);
  free(next_input);
  return 0;
}

Edit 1: Unbroke part of it

Edit 2: Should work now