Say that you're playing Sonic 3 and Knuckles and you've got most of the Super Emeralds. Maybe you've got one left to do, but you've messed up all the Giant Ring challenges, or otherwise don't have any more to go.
When you clear the game, will those Giant Rings respawn? Or do you have to start all over again?
I knew the answer by experience (I cleared the game like 15 times), but I just cleared the game today to be able to give you a more detailed answer.
When you clear the game, the save data you used turns into a "Clear Data", as shown in the image below:
Notice the missing Hyper Emeralds on the image. You may cycle through the zones using this very save file (by pressing up and down). This way you can choose a Sonic & Knuckles' zone to retrive the missing emeralds, because the giant rings respawn when o pick a zone on your "Clear Data", as seen on the image below:
Be aware. If you want to get Hyper Emeralds you must play the Sonic & Knuckles' zones (any zone from Mushroom Hill Zone onwards), because if you play a Sonic 3 zone (from Angel Island to Lauch Base) all giant rings there are regular ones and do not lead you to the room where the Hyper Emerald Stages are, as seen on the image below. Instead, they give you 50 rings.
Finally, the game saves automatically after you get a new Hyper Emerald. I just tested the following scenario: I entered the Mushroom Hill Zone, got a new Hyper Emerald, turned off and on the console, and my save file had added the Hyper Emerald I just got. Plus, I entered Mushroom Hill Zone again, went to the very same Giant Ring I used to get the last Hyper Emerald and the Giant Ring had respawned again. This happens only on cleared data, though. Giant Rings do not respawn if your data isn't cleared.
They do not respawn during a first playthrough. You need to complete the game, then when you load a level on the same save file, you can attempt to complete the Emeralds you missed as covered by @Lucas.
Note that while Lucas' answer holds true for Sonic 3 (base game) and Sonic 3 & Knuckles (joined game), it is impossible to view and load a save game in Sonic & Knuckles, due to this cartridge lacking the save/load features. (This functionality is handled by the Sonic 3 cartridge when you join them).
Thus if you are playing S&K un-joined it is impossible to get to a scenario where Giant Rings respawn.
