1. Stating the problem: We want to find the function $g\circ g \circ g (x)$ where $g(x) = 1 + x^2$. 2. Start by finding $g(g(x))$: Substitute $x$ in $g(x)$ with $g(x)$ itself. $$g(g(x)) = 1 + (g(x))^2 = 1 + (1 + x^2)^2$$ 3. Expand $g(g(x))$: $$g(g(x)) = 1 + (1 + 2x^2 + x^4) = 1 + 1 + 2x^2 + x^4 = 2 + 2x^2 + x^4$$ 4. Next, find $g(g(g(x)))$: Substitute $x$ in $g(x)$ with $g(g(x))$. $$g(g(g(x))) = 1 + (g(g(x)))^2 = 1 + (2 + 2x^2 + x^4)^2$$ 5. Expand the square: $$(2 + 2x^2 + x^4)^2 = 2^2 + 2 \cdot 2 \cdot 2x^2 + 2 \cdot 2 \cdot x^4 + (2x^2)^2 + 2 \cdot 2x^2 \cdot x^4 + (x^4)^2$$ More explicitly: $$= 4 + 8x^2 + 4x^4 + 4x^4 + 4x^6 + x^8 = 4 + 8x^2 + 8x^4 + 4x^6 + x^8$$ 6. Therefore, $$g(g(g(x))) = 1 + 4 + 8x^2 + 8x^4 + 4x^6 + x^8 = 5 + 8x^2 + 8x^4 + 4x^6 + x^8$$ Final answer: $$\boxed{g(g(g(x))) = 5 + 8x^2 + 8x^4 + 4x^6 + x^8}$$