say for example we have the distance traveled by a vehicle as a function of time. if the speed(change in distance) is constant then this would be a linear function of order 1. if there was acceleration(change in (change in distance)) then this would be an order of 2. and if the acceleration itself was changing then this would be an order of 3.
my question is what would a function of order 4 be like? is it change in the change in acceleration and is it achievable in real life? also whats a function of order 1/2 be like?
What you're talking about is the $n^{th}$ derivative of the position vector with respect to time.
The fourth derivative of position with respect to time is jounce.
As for the half derivative, you can read more about it here, but I don't think it has much physical significance like velocity, acceleration or jounce.