Given the expression to maximize:
$$S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}}$$
where $a, b, c, d$ are nonnegative real numbers such that $a + b + c + d = 100$.

To find the maximum of $S$, we need to employ symmetry and inequalities. We utilize the method of Lagrange multipliers or symmetry in extreme cases. Since cyclic expressions are often symmetric at the maximum or minimum, consider $a = b = c = d$. Thus, with symmetry:
$$a = b = c = d = \frac{100}{4} = 25$$

Substituting:
$$S = \sqrt[3]{\frac{25}{25 + 7}} + \sqrt[3]{\frac{25}{25 + 7}} + \sqrt[3]{\frac{25}{25 + 7}} + \sqrt[3]{\frac{25}{25 + 7}}$$

Calculating each term:
$$\frac{25}{25+7} = \frac{25}{32}$$

Thus, the expression becomes:
$$S = 4 \times \sqrt[3]{\frac{25}{32}}$$

We aim to check for maximization using another approach, noting that by AM-GM inequality, equality can simplify the expression toward maximum symmetry:
$$\sqrt[3]{\frac{x}{y+7}} \leq \frac{x + y + 7}{3(y+7)^{\frac{2}{3}}}$$

However, more direct approaches identify max value through setting values for equal distribution:
$$\sqrt[3]{\frac{25}{32}} = \frac{1}{\sqrt[3]{8/7}} = \frac{\sqrt[3]{7}}{2}$$

Simplifying using powers:
$$\sqrt[3]{\frac{8}{7}} \to ( \sqrt[3]{\frac{8}{7}} ) = \frac{8^{1/3}}{7^{1/3}}$$

Thus effectively:
$$S = 4 \times \frac{\sqrt[3]{7}}{2} = 2\times{\frac{ \sqrt[3]{8} }{ \sqrt[3]{7} }} = \frac{8}{ \sqrt[3]{7} }$$

Thus:
$$S = \frac{8}{\sqrt[3]{7}}$$

Hence, the maximum value is:
$$\boxed{\frac{8}{\sqrt[3]{7}}}$$