This is a wiki to provide a list of plugins and extensions that the community feel are essential for them to be able to use Classic Commerce. Some discussion may also have taken place here: https://github.com/ClassicPress-research/classic-commerce/issues/8
Stripe - currently the WooCommerce Stripe Payment Gateway plugin works with WP 4.4 or higher, so it should work with ClassicPress. Whether it continues to work with WC 3.5.3 will need monitoring.
Paypal - Note there is a basic Paypal payment gateway included with the core installation. The WooCommerce PayPal Gateway plugin currently works with WP 3.3 or higher, so it should also work with ClassicPress. Same with WooCommerce PayPal Pro Payment Gateway. Whether these continue to work with WC 3.5.3 will need monitoring.
VAT ID -This is the registered number that you can fill in on the ordering form so that VAT is not charged (if your number is valid). This is something that you MUST have with an e-commerce site in Europe. For example: https://woocommerce.com/products/eu-vat-number/
Name your Price
Facebook integration - such as https://woocommerce.com/products/facebook/
Print shipping labels
Square integration - https://woocommerce.com/products/square/