Execute jQuery function after another function completes

You should use a callback parameter:

function Typer(callback)
{
    var srcText = 'EXAMPLE ';
    var i = 0;
    var result = srcText[i];
    var interval = setInterval(function() {
        if(i == srcText.length - 1) {
            clearInterval(interval);
            callback();
            return;
        }
        i++;
        result += srcText[i].replace("\n", "<br />");
        $("#message").html(result);
    },
    100);
    return true;


}

function playBGM () {
    alert("Play BGM function");
    $('#bgm').get(0).play();
}

Typer(function () {
    playBGM();
});

// or one-liner: Typer(playBGM);

So, you pass a function as parameter (callback) that will be called in that if before return.

Also, this is a good article about callbacks.

function Typer(callback)
{
    var srcText = 'EXAMPLE ';
    var i = 0;
    var result = srcText[i];
    var interval = setInterval(function() {
        if(i == srcText.length - 1) {
            clearInterval(interval);
            callback();
            return;
        }
        i++;
        result += srcText[i].replace("\n", "<br />");
        $("#message").html(result);
    },
    100);
    return true;
        
    
}

function playBGM () {
    alert("Play BGM function");
    $('#bgm').get(0).play();
}

Typer(function () {
    playBGM();
});

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/1.7.2/jquery.min.js"></script>
<div id="message">
</div>
<audio id="bgm" src="http://www.freesfx.co.uk/rx2/mp3s/9/10780_1381246351.mp3">
</audio>

JSFIDDLE

You can use below code

$.when( Typer() ).done(function() {
       playBGM();
});