Try

elgg.my_plugin.init = my_function()

Doesn't work, it says the function isn't defined.

Sure, you should provide it:

elgg.provide('elgg.my_plugin');

But for Elgg 2.0 it's not recommend (use modules instead of)

I don't wanna split my JS into multiple modules, I prefer to have them in one file.

I prefer to have them in one file

That's what I'm doing on our projects too ;) Just use the above examples.

Yeah I'm just gonna have to use regular jQuery for that one specific function lol.

Any particular reason you don't want to use modules?

I'd have to learn a whole new structure, which would hinder my new theme development which involves 99% ajax. I'll eventually learn it one day.

Here's our snippet which makes our websites 100% AJAXify.

We use HistoryJS for it and combine different methods.

For async load HistoryJS we use definition function:

elgg_define_js('history', array(
        'src' => "mod/theme/vendors/js/jquery.history",
        'deps' => array('jquery')
    ));

For initialize history's tools/functions we provide it into views path:

elgg.provide('elgg.history');

elgg.history.init = function() {

    $(window).bind('statechange', function() {
        require(['history'], function() {
            var state = History.getState();

            if (state && elgg.trigger_hook('history', 'statechange', state, true)) {
                elgg.history.get_page({
                    url: elgg.normalize_url(decodeURIComponent(window.location.href)),
                    forward: state.data.forward ? true : false
                });
            }
        });
    });

//.....etc

}

Nice! If you don't mind, I want you to check out my new theme once it's done. Here's a preview of the landing page: https://drive.google.com/file/d/0BzzvL4ngEoyfQTY2ZXpOSmVLVUk/view?usp=sharing