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Thanks to everyone who's commented! Based on all your answers I was able to come up with this which works perfectly:

drupal_add_js( array( 'themepath' => base_path() . path_to_theme() ), 'setting' );

Which you can then reference in your .js file as

var THEMEPATH; //global variable
$(function(){
    THEMEPATH = Drupal.settings.themepath;
})

//THEMEPATH has to be a global variable or else it won't work

So in the end, you simply use it in jQuery as: THEMEPATH + '/code/test-ajax.php'