I like Dan Clarke's answer a lot. I will provide a similar one.
The most common pet turtle in the USA is the red-eared slider.
These buggers are usually 20 centimeters long. That isn't even ten inches. Now, some mad scientists have done science to them. Here's a paper on them:
Microstructure and mechanical property of turtle shell
And it says:
(...) They measured the mechanical properties of the bone fibers by using a nano indenter with the results of elastic modulus ∼20 GPa and hardness ∼1 GPa.
Which is within the same order of magnitude observed with the tooth of a limpet.
The authors mention that the scutes used in the experiment have a thickness ranging from 2.8 to 4 millimeters (~0.11 inches - 0.15 inches). Now, one thing that is a trope related to dragons is that they are obscenely large. Assuming that dragon scales are made of the same material as turtle shell scutes, and that thickness has a linear* relationship to body length... A dragon the size of a Tyrannosaurus Rex (~12 meters) would have their thickest scales (I suppose around the middle of the back) being around 24 centimeters thick. That is almost 10 inches. Ten expletive inches of turtle shell-hard material separating the atmosphere from their subcutaneous fat. Imagine trying to put an arrow through that.
But let's not stop there.
Smaug was at least 20 meters long, so his thickest scales would be 40 centimeters thick (~1.31 feet).
I remember something about chromatic dragons from D&D reaching 30 meters long sizes. Their thickest scales would be 60 centimeters thick (~2 feet).
And don't get me started on the really big beasts such as D&D's Tiamat, Magic: the Gathering's Nicol Bolas and Shadowrun's Mountainshadow, which all would have their thickest scales being measured in multiple meters.
* square would be more reallistic, but the beast would be too thick to be believable.
If you think that those scales would be too thick, making for a ridiculous creature that is more scales than anything else in mass in volume, you can taylor the scales down so that each is the approximate size of a red eared slider (as has been suggested in the comments). They would still provide an amazing level of protection. See Valthek's answer - the disposition of the scales may play a more important role than their actual thickness.