New Scientist has a list of 13 things that don't make sense, starting with the Placebo effect.
This reminds me of the famous lecture by Lord Kelvin in 1900 where he said there were just two problems left to solve in Physics.
As I recall it, his advice to students was not to go into Physics as Newton's Laws of Motion and the Theory of Thermodynamics had pretty much explained everything - apart from these two minor issues - but I couldn't find any reference specifically giving that advice on the Internet.
The two problems were to do with detecting the ether (the medium through which it was assumed electro-magnetic radiation such as light was propagated) and describing the radiation emitted by a black body.
In seeking the solutions to these, physicists came up with the Theory of Relativity to describe how gravity works and the Laws of Quantum Mechanics to describe how sub-atomic particles (such as electrons) behave. In other words, the whole of modern physics.