You sound a bit like me, rarely taking painkillers unless you absolutely need to. However if you’re unfamiliar with painkillers, it would be good to ask your medical team before discharge for a plan for your pain relief. Generally, it’s muvh better to control pain before it peaks, so taking the permitted four doses of paracetamol per day, even if you’re not in agony when the next dose is due, is better and more effective than letting it all wear off, allowing the pain to peak and then trying to battle it back down. Paracetamol is a very effective painkiller, usually under-rated as it’s so familiar and so easily available, but also because people don’t take it properly, instead holding out until it’s agony.
You have no way of predicting exactly what kind of shape you’ll be in, but it’s safe to say that even the smoothest labours are physically arduous and place huge demands on your body. I fear that if you plan to avoid painkillers, you might be setting yourself up for disappointment. No need to stock up on bucketfuls if you have easy access (by you or someone else) to the shops but it would be prudent to have a small quantity of paracetamol and ibuprofen to hand in case you need it at a time when shops are closed.
Also birth injuries aren’t the only source of pain as people have already said - starting breastfeeding can give you awfully painful blisters or cracks if unlucky, breast engorgement can be very painful, complications like mastitis need something to control your fever etc.
I don’t mean to frighten you but I do think it unlikely to avoid any possible need for pain relief in the early days. And please don’t forget that your comfort and pain matter too, so if you need something, take it. And if that’s not working, ask your doctor or midwife.
Final thing - the early days with a baby can be a confusing blend of night and day and it can be easy to lose track of what medication you have taken when. I had a notepad and pen taped inside the cupboard where I kept medicines and would write down every dose. Some people send themselves a text message, etc. Think about what might work for you and set it up in advance to keep track.
Best wishes for a safe healthy delivery and recovery, whatever happens.