"Avoid chemical drain cleaners entirely. They rarely work on serious toilet clogs and can damage pipes or seals."
Couldn't agree more with this—those chemical cleaners are usually more trouble than they're worth. If the auger isn't cutting it, you might want to try dish soap combined with hot water. Let it sit for a good 15-20 minutes before plunging again. Had a similar stubborn clog once, and this trick finally did it after multiple auger attempts... worth a shot before calling in backup.
Yeah, dish soap and hot water can definitely help loosen things up, but I'd be careful with the water temperature—too hot and you risk cracking the porcelain. Warm water usually does the trick without the risk. Had a clog once that wouldn't budge even after plunging and augering multiple times. Ended up pouring in some baking soda and vinegar, let it fizz for about half an hour, then plunged again. Worked surprisingly well... and no harsh chemicals involved.