Is it possible to query the result of a query?

Absolutely. Since you can instantiate to a variable a query you can do it as much times as you want. Of course query loops require a lot of connections to the database so you have to be careful. Very often if you need many subqueries it basically means that your workflow can be enhanced in some way.

Example of a subquery:

$some_posts = new WP_Query();
$some_posts->query('cat=1&showposts=2');

while ($some_posts->have_posts()) : $some_posts->the_post();

    // Start new subquery with query values
    $inner_query = new WP_Query();
    $inner_query->query('cat=1&showposts=2');

    while ($inner_query->have_posts()) : $inner_query->the_post();

        // Do something in your inner/sub query

    endwhile;

endwhile;