EulerNumberGenerator.cs

using System;

namespace EulerNumberGenerator
{
    class Program
    {
        static Random rand = new Random();

        static void Main(string[] args)
        {
            double sum = 0;
            int amount = int.MaxValue;
            for (int i = 0; i < amount; i++)
            {
                int picks = DoRandomPicks();
                sum = sum + picks;
                double avg = sum / (i + 1);
                Console.WriteLine("Avg:" + avg + " Error: " + (avg - Math.E));
            }

        }

        static int DoRandomPicks()
        {
            int pickCounter = 0;
            double sum = 0;
            while (sum < 1)
            {
                double r = rand.NextDouble();
                sum = sum + r;
                pickCounter++;
            }
            return pickCounter;
        }
    }
}

Sample output:

Avg:2.70660032651792 Distance:-0.0116815019411254
Avg:2.70662313432836 Distance:-0.0116586941306869
Avg:2.70664593859308 Distance:-0.0116358898659632
Avg:2.70666873931292 Distance:-0.0116130891461275
Avg:2.70669153648869 Distance:-0.0115902919703532