Wordpress - force enqueue script to be first in order of prominence

You just need to enqueue your scripts before plugin does it. You can do it by setting priority to 0 for your hook. For example, do the following:

add_filter( 'wp_enqueue_scripts', 'wpse8170_enqueue_my_scripts', 0 );
// or if you enqueue your scripts on init action
// add_action( 'init', 'wpse8170_enqueue_my_scripts', 0 );

function wpse8170_enqueue_my_scripts() {
    wp_enqueue_script( 'myscript', 'http://path/to/my/script.js', array( 'jquery' ) );
    // my else scripts go here...

Setting up priority for your hooks will put it to the beginning of calling queue and your scripts will be added first.