How would you increase the maximum size of a players hand

You would increase the value for the integer MaxHandSize.

 static void Main(string[] args)
        {
            List<String> AllowedWords = new List<string>();
            Dictionary<Char, int> TileDictionary = new Dictionary<char, int>();
            int MaxHandSize = 20;

For example you could change it to 25:

 static void Main(string[] args)
        {
            List<String> AllowedWords = new List<string>();
            Dictionary<Char, int> TileDictionary = new Dictionary<char, int>();
            int MaxHandSize = 25;