Since album_photos has a mapping relationship between photos and albumns, you can specify the number of photos to join on by using TOP:
SELECT a.album_name, a.album_id, p.*
FROM albums a
INNER JOIN album_photos ap ON
ap.photo_id = (select top 5 photo_id from album_photos where a.album_id = ap.album_id order by photo_id)
INNER JOIN photos p ON ap.photo_id = p.photo_id
The Order by photo_id in the subquery will ensure the same 5 (or fewer) photos are returned
EDIT PER COMMENT. Modifying to use MySql LIMIT instead of T-SQL TOP
SELECT a.album_name, a.album_id, p.*
FROM albums a
INNER JOIN album_photos ap ON
ap.photo_id = (select photo_id from album_photos where a.album_id = ap.album_id order by photo_id LIMIT 0, 5)
INNER JOIN photos p ON ap.photo_id = p.photo_id