How can I retrieve data from a database using asp.net and then feed that data into a javascript array?

From your question, it sounds like you are working with ASP.NET web forms. If that is the case, you will probably want to retrieve your 15 values using an AJAX call. If you build the JS synchronously in the Page_Load, it will be easy to view source and see what is going on... That said, anybody can turn on Network capture and see the hangman values coming back in the AJAX response.

Maybe consider using a page method. You would make a static web method in your code behind. Here is what pseudo code would look like with C# and jQuery.

On the server in your code behind (say it is hangman.aspx.cs)

[WebMethod]
[ScriptMethod]
public static string[] GetHangmanValues()
{
    string[] values = new string[15];
    //retrieve your hangman values from the db and return them
    return values;
}

On the client side:

var HANGMAN_APP = {};
$(function () {
    $.ajax({
        type: "POST",
        dataType: "json",
        contentType: "application/json",
        url: "hangman.aspx/GetHangmanValues",
        data: "{}",
        success: function (data) {
            //Microsoft page methods wrap the data in data.d
            //use data.d to populate HANGMAN_APP
        },
        error: function () {
            alert("Error calling the page method.");
        }
    });
});

This way of doing it will keep the values from being displayed in page source, but I just want to reiterate: Network traffic and/or a javascript debugger would not protect your values if somebody really wanted to see them. Also, an extra benefit is that Page Methods don't go through the entire page life cycle or deal with ViewState (like UpdatePanels do), which makes them very efficient for this sort of thing.