I have 2 payment gateways integrated with my shop: 2checkout and Paypal Express checkout.
I would like to add another payment provider. For example (and just for example) Stripe.
Freddie here from the Shopify Support team, happy to help.
You can certainly add a third payment gateway, but it will need to be another external payment gateway where the customer is redirected to the payment gateway's checkout to complete the purchase, such as what you see with PayPal Express Checkout.
You can have one direct payment gateway set up, such as 2Checkout or Stripe, then you can have PayPal Express Checkout, then you can also have an additional option that is an external payment gateway. You can enable this by going to Settings > Payments, and you'll have this option in the Alternative Payments section to enable your third external option.
For a full list of which payment gateways are direct or external, you can click right here.
I hope this information is helpful! Let me know if you have any further questions that I can help you with.
Shopify Support - firstname.lastname@example.org