To find the value of the expression, we need to first simplify the expression by using the properties of trigonometric functions.

Given:

arccos(1) = 0 (since cos(0) = 1)
arctg(-√3/3) = -π/6 (since tan(-π/6) = -√3/3)

Substitute these values into the expression:

Ctg(arccos(1) + 2arctg(-√3/3))
= Ctg(0 + 2(-π/6))
= Ctg(-π/3)

Since cotangent is the reciprocal of tangent, we have:

Ctg(-π/3) = 1/tan(-π/3) = 1/(√3)
= √3/3

Therefore, Ctg(arccos(1) + 2arctg(-√3/3)) = √3/3.