There's a bunch of pain relieving medications without Tylenol, so I'll give you a pretty big list, but keep in mind that some of them might be completely inappropriate for whatever pain it is you've got. Your physician will know which one might work best for you.
Opiates like codeine (hydrocodone and oxycodone) are usually compounded with another medicine, like the acetaminophen you mentioned, but also with aspirin. Among the medicines that codeine is mixed with aspirin are Lortab and Percodan, and they are short-acting. Opiates like morphine (hydromorphone) that are not compounded include the drug Dilaudid.
Some other analgesics without Tylenol include Darvon and Ultram or Tramadol.
There are anti-seizure drugs that are also good nerve medicines, they are not combined with Tylenol, and they are given to people who have fibromyalgia-like pain, for example. Among those nerve drugs are Neurontin or Lyrica, Topamax, Tegretol, and Dilantin.
So, I imagine you could take any one of those medicines and get the pain relief you need without also having to take Tylenol, but as to short-term or long-term effect, or whether they are too strong or not strong enough, or whether any of them have liver damage as a possible side effect, that would be something your prescribing physician would know or could find out for you.
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