Stonehenge

I think the evidence for when and how Stonehenge was built is pretty convincing. How the stones were transported is more problematic.

I suspect that if the stones were transported using a boat of some kind, then some were probably lost, and may be discovered by marine/underwater archaeologists at some point in the future. It is even possible (though much less likely) that organic remains of parts of the equipment used to move the stones could be recovered from anaerobic deposits in marshy areas along the transport route. Other than that, experimental archaeology probably remains our best hope for finding ways that they could have been moved.

As to who built it and why, I suspect that the questions are related. If we knew who built it, we would probably understand why, and vice-versa. A lot of large monuments were constructed during the Neolithic - long barrows, causewayed enclosures, Silbury Hill, and henges. These all would have had a purpose within those societies, but I can't help thinking that large communal projects like these also help forge a sense of community and ownership ("this land belongs to us"). Ancient peoples weren't stupid!

And, let's be honest, large projects like these would also have provided a great way to keep young men (and, very possibly, young women as well) occupied and out of trouble at quiet times in the agricultural year for decades!