You can go with two queries, the first to get your taxonomies like this :
$categories = get_categories();
More infos about the params here.
And then browse your category collections and get the first n posts of it like this :
foreach($categories as $cat)
{
//get the $n posts from this category
$post_query = new WP_Query(
array('post_type' => 'post',
'posts_per_page' => $n ,
'tax_query' => array(
array(
'taxonomy' => $cat->taxonomy,
'field' => 'slug',
'terms' => $cat->term_id,
)
)
)
);
echo '<h2>'.$cat->name.'</h2>';
while($post_query->have_posts()){
$post_query->the_post();
echo get_the_title();
}wp_reset_query();
}
Hope it will help =)