Pharmacology lesson (I'm an anaesthetist so pain control is one of my main goals!)
Best thing to do is combine drugs with different mechanisms of action then you will get combined effect. The main groups are broadly non-steroidal anti-inflammatory drugs (NSAIDS), paracetamol, and opioids.
NSAIDS (aspirin, ibuprofen, diclofenac, etc.) - Broadly they reduce the body's inflammatory response and decrease tissue swelling round trauma. These are good painkillers when used in combination with opiates or paracetamol, but not that effective used seperately. They tend to be more effective when prescribed regularly for a few days rather than just as required. Side-effects include increased bleeding, renal impairment, and gastric ulceration. Aspirin generally causes most bleeding side-effects of the drugs mentioned, while ibuprofen and diclofenac tend to be better painkillers.
Paracetamol - Mechanism of action is poorly understood, but has anti-inflammatory properties. Few side-effects. Anadin/Hedex/etc. tend to have reduced dose paracetamol and caffeine in them to somehow justify the extra cost. Frankly it isn't worth the extra money! Just get the cheapest paracetamol you can find.
Opioids (codeine, tramadol, morphine, diamorphine, fentanyl, etc.) - The big boys, which work on receptors in the nerves and brain! Codeine is available over the counter in the UK at a strength of 8mg per tablet, but 30mg per tablet codeine is available by prescription. It also comes as co-codamol which is combination of paracetamol and codeine.
The "best" painkiller combination you can get over the counter is (bear in mind these are adult doses and this is in no way suitable for all due to side-effects, contraindications, and cautions!)
Co-codamol (1g paracetamol with 16mg codeine) four times daily
Ibuprofen 600mg three times daily
Personally if I take painkillers I use ibuprofen AND paracetamol as a first line treatment. Might as well use something that's likely to work first time!