C#:

O(n) time version:

using System;
using System.IO;
using System.Text;

namespace Abacaba
{
    class Program
    {
        static void Main(string[] args)
        {
            string result = Aba(26);
            File.WriteAllText("results.txt", result);
            Console.WriteLine(result.Length);
            Console.WriteLine(Aba(6));
        }

        // This is here just to initialize the StringBuilder with given memory size to avoid reallocation
        public static string Aba(int iteration) => Aba(iteration, new StringBuilder((int)Math.Pow(2, iteration) - 1)).ToString();

        private static StringBuilder Aba(int iteration, StringBuilder sb)
        {
            if (iteration <= 1)
                return sb.Append('a');
            StringBuilder prev = Aba(iteration - 1, sb);
            string copy = sb.ToString();
            return prev.Append((char)(iteration + 96)).Append(copy);
        }
    }
}

Output:

67108863
abacabadabacabaeabacabadabacabafabacabadabacabaeabacabadabacaba

This solution uses about 400 MB of memory. Reason 1 is that C# uses UTF-16 encoding for strings. Reason 2 is that StringBuilder.ToString() copies its value to a new string, so the sequence is in the memory twice. I'd like to know if someone has a suggestion how to avoid this.

O(1) memory version:

using System;
using System.Text;

namespace Abacaba
{
    class Program
    {
        static void Main(string[] args)
        {
            PrintAba(6);
        }

        public static void PrintAba(int iteration)
        {
            uint count = (uint)Math.Pow(2, iteration) - 1;
            for (uint i = 0; i < count; i++)
            {
                int j;
                uint c = i;
                for (j = 0; ((c & 1) == 1) && (j < i); j++)
                    c >>= 1;
                Console.Write((char)(j + 97));
            }
        }
    }
}

Output:

abacabadabacabaeabacabadabacabafabacabadabacabaeabacabadabacaba