Limiting the amount of posts this displays

You have posts_per_page = -1, which means show unlimited posts. Change the -1 to the number of posts you want to appear.

Also you have output that skips if your “next_open_day” meta value is null, but the query will only run 3 times, so you need to move that logic into the query itself by using these arguments in the query_posts().

<?php
$metakey = 'next_open_day';
$args = array('post_type' => 'page',
              'post_parent' => 2,
               'posts_per_page' => 3,
               'orderby' => 'title',
               'order' => 'ASC',
               'meta_query' => array( 
                   array('key' => $metakey,
                         'value' => '',
                         'compare' => '!='
                   )
               )
);

query_posts($args);
?>