Taking $y = f(x)$ in the given equation, we get
$$f(x) + f(f(f(x))) + f(f(f(x))) = x + f(x),$$
from which, by expressing $f(f(f(x)))$, we find
$$f(f(f(x))) = \frac{x}{2}. \qquad (7)$$
Taking $y = 0$ in the given equation, we get $f(x) + f(f(0)) + f(f(f(x))) = x + 0$; using the equality (7) here, we get after simplification
$$f(x) = \frac{x}{2} - f(f(0)). \qquad (8)$$
Substituting $x = f(f(0))$ into equation (8) and applying equation (7) on the left hand side, we get
$$\frac{0}{2} = \frac{f(f(0))}{2} - f(f(0)), \qquad (9)$$
which after simplification gives $f(f(0)) = 0$. Consequently, equation (8) takes the form $f(x) = \frac{x}{2}$. But in this case, $f(f(f(x))) = \frac{x}{8}$, whereas from equation (7) $f(f(f(x))) = \frac{x}{2}$. Since in general $\frac{x}{2} \ne \frac{x}{8}$, there are no functions that satisfy the given functional equation.