Welcome to the Shopify Community! Simply put, through the natural Shopify settings it's not possible for a customer to use more than one discount in any given order. I understand that offering your customers multiple discounts can be important, especially in big sale periods. So, I've gone ahead and submitted some feedback on your behalf to our developers about requesting this feature for the future. While I can't guarantee any changes, the more feedback we receive, the better!
That being said, in order to offer more than one discount to a customer in a single order, you have a couple options:
Use the Compare at price (sale price) feature to set the products as on sale (% discount). Then, change the pricing back once the sale is over. Since this isn't using the discount feature, customers can then still use a discount code, or automatic discount on top of a sale price.
Use an app from the Shopify App Store to be able to offer more than one discount at a time. Many apps exist that allow your customers to use more than one discount in a single order. Try searching "stack discounts" in the App Store to find some options.
If you decide to go with the first option (using the compare at feature) I'd recommend using the Bulk Editor to make the changes, since this will be the fastest way to change the pricing on multiple products.
Let us know how this goes and if you have any other questions!
Hilary | Community Moderator @ Shopify - Was my reply helpful? Click Like to let me know! - Was your question answered? Mark it as an Accepted Solution - To learn more visit the Shopify Help Center or the Shopify Blog