Hi,
There is no limit per se. However, the column badge_product_id where the ids of the products are stored in by default limited to 255 characters for performance issues.
If you want to store more product ids, you can edit that column of the hikashop_badge table via your phpmyadmin and change the column type to "text". The limit will then be around 65000 characters.
However, it might be better to actually create an unpublished category and select that category on your badge and then add that category as a second category to the products on which you want the badge.