Time it takes to reach maximum current?
Is the same applied for singular wires(not forming a loop/multiple
  turn loop)?


The example you ask for consideration is an imposibility. If a wire passes current then it MUST have a return path to the signal source and this forms a loop. Even a monopole antenna passes current and has a return path even though physically it looks like a wire with a connector at one end.

Having made my rant, you can theoretically predict that a wire does have self-inductance and that inductance is maximum when the "return path" is a gazillion miles away: -

http://latex.eeweb.com/eq.latex?L%3D2l%5Cleft%20(%20%5Cln%20%5Cleft%20(%20%5Cleft%20(%20%5Cfrac%7B2l%7D%7Bd%7D%20%5Cright%20)%5Cleft%20(%201%2B%5Csqrt%7B1+%5Cleft%20(%20%5Cfrac%7Bd%7D%7B2l%7D%20%5Cright%20)%5E%7B2%7D%7D%20%5Cright%20)%20%5Cright%20)-%5Csqrt%7B1+%5Cleft%20(%20%5Cfrac%7Bd%7D%7B2l%7D%20%5Cright%20)%5E%7B2%7D%7D+%5Cfrac%7B%5Cmu%20%7D%7B4%7D+%5Cleft%20(%20%5Cfrac%7Bd%7D%7B2l%7D%20%5Cright%20)%20%5Cright%20)

Where "l" is the wire length and "d" is the diameter.


  From the equation above, that's the time it takes for current to reach
  2/3 it's value?


No, it's not two thirds it's 0.63212055882 or \$1-e^{-1}\$