What is the best medication for continuous pain relief?

It depends on your child (what medication that child responds best to) and on the type of pain. Some children will respond better to acetaminophen (Tylenol) than they will to ibuprofen (Advil or Motrin). For others it is the reverse.

Prostaglandins are chemicals (unsaturated carboxylic acids) in the body that help it activate and transmit pain signals, among many other things. Prostaglandins are usually good things - warning the brain that something is wrong. Acetaminophen is thought to work on COX-3, in theory reducing those specific type of prostaglandins. Ibuprofen is thought to do the safe, but with COX-2.

Ibuprofen seems to be best indicated for situations where inflammation is of especially great concern (eg, a sprained ankle, migraine). It is always a good idea to ask your doctor, however. Some doctors will advise to alternate doses of acetaminophen with ibuprofen every three to four hours.

Just know that you should not be giving any medication that contains aspirin to a child under 18 years of age, unless your doctor directs you otherwise. This means you need to read the label of active ingredients on any medication, because while certain medications are obviously aspirin (Bayer, Bufferin, St. Joseph's), there are other medications (namely Excedrin brand, or a CVS/Walgreens/Rite-Aid Excedrin copycat) which contains aspirin as well as acetaminophen.