Is it possible to retrieve past order receipts?
I may have neglected keeping track of all of the purchases that I've made, and I thought "Surely someone in the forums might have the answer to this question".

Pretty sure I deleted most of them out of sheer misguided temperament.

On the GOG website (don't know about galaxy) Hold your mouse over your username at the top, and then click on "orders and settings"

To add:
Hover your mouse over the header where there is

ORDER #XXXXXXXXX
date

Then it shows a drop down arrow where you can view the order specific receipt.

Galaxy has it in the sidebar called "Orders history". On the right side the down arrow is always visible which brings up "View Receipt" and "Contact support". It's searchable (which works and brings up the orders containing the provided phrase in any game title). There sadly is no way to increase the number of items on a page, and the page navigation is on the bottom only. It also has a filter dropdown called "Orders History", where one can click on various things that seems to be broken so that one item does what the previous item says, and also each itrem is a toggle, but does not reflect its state. Clicking on the sidebars "Orders History" clears all filters.