You know how things you can interact with have a blue aura when you get near them? But sometimes you open up the menu and it's all greyed out because you already dug that peat bog today, or you harvested the tree _and_ it's been watered and petted recently. If the aura were a different colour, you'd know you couldn't interact with it, and you wouldn't spend time waiting for the menu to open.
Only snag with that is that you can rename animals as much as you like, as far as I can tell, so you'd still have a live option in the menu for those even after you've maxed everything else. But it might help save time and server cycles for other objects.
I'd still want to be able to open the menu even if I couldn't interact, just so that I can see what options might be available later and why they're inactive now. So just disabling the menu or replacing it with a message wouldn't work.