In many contexts, I have seen phrases like "... function $f$ is twice differentiable ...". I can understand that the function $f$ can be differentiated twice or in other words its second derivative exists. But does this mean third derivative (or higher) does not exist or is equal to $0$?
In the same way, what would we call a function like $e^x, \sin x, \dots$ which can be differentiated any number of times?
Twice differentiable functions can be thrice differentiable.
Functions like $e^{x}, \sin x $ are infinitely differentiable functions or $C^{\infty}$ functions..