= Solution
Use $\mathbb T^n=(\mathbb R/2\pi\mathbb Z)^n$ and the uniform metric $d(f,g)=\|f-g\|_\infty$. Let $N(\varepsilon,E)$ be the least number of radius-$\varepsilon$ balls in this metric covering $E$. The <Kolmogorov entropy>, equivalently <metric entropy>, is
$$
\boxed{H(\varepsilon,E)=\log N(\varepsilon,E)}.
$$
It is infinite if no finite cover exists; set the entropy of the empty set to zero. Changing the logarithm base or requiring centres in $E$ changes only the constants and resolution factors in the estimates below.
For an explicit appropriate norm, take
$$
\|f\|_{C^p}=\max_{|\alpha|\le p}\|\partial^\alpha f\|_\infty.
$$
An equivalent standard norm gives the same exponents, with different constants. We prove the two bounds for the <entropy of a smooth periodic function ball> separately.
For the lower bound, choose a nonnegative <bump function> $\varphi\in C_c^\infty((-1,1)^n)$ with $0\le\varphi\le1$ and $\varphi(0)=1$. Let $K$ bound all its derivatives of orders at most $p$, and choose $c>0$ with $cK\le1$. For sufficiently small $h\le1$, place $L\ge c_nh^{-n}$ translates of its scaled support disjointly on the <torus>, for instance by taking grid centres separated by $3h$ in a fundamental cube. Define the real functions
$$
f_\sigma(x)=c h^p\sum_{j=1}^{L}\sigma_j\varphi\left(\frac{x-x_j}{h}\right),
\qquad \sigma\in\{0,1\}^{L},
$$
using periodic extension. Disjointness means that at any point only one summand contributes. For every $|\alpha|\le p$, its derivative is bounded by $c h^{p-|\alpha|}K\le1$. Thus every $f_\sigma$ lies in $B_{n,p}$. If $\sigma\ne\tau$, evaluate at a centre where the binary coefficients differ to obtain
$$
\|f_\sigma-f_\tau\|_\infty=ch^p.
$$
Choose $h=(4\varepsilon/c)^{1/p}$ for sufficiently small $\varepsilon$. The $2^L$ functions are more than $2\varepsilon$ apart, so no radius-$\varepsilon$ ball covers two of them. The <metric covering number> consequently satisfies
$$
H(\varepsilon,B_{n,p})\ge L\log2\ge c'_{n,p}\varepsilon^{-n/p}.
$$
To include the remaining $\varepsilon<1/2$, observe that the two constant functions $1,-1$ belong to the ball and require distinct covering balls. Reducing the positive lower-bound constant extends the estimate over that fixed remaining interval.
For the upper bound, use this precise version of <multivariable Jackson approximation>: for every $f\in C^p(\mathbb T^n)$ and integer $N\ge1$, there exists a real <trigonometric polynomial> $P_N$ with $|k_j|\le N$ in every coordinate and
$$
\|f-P_N\|_\infty\le J_{n,p}N^{-p}\|f\|_{C^p}.
$$
Choose $N=\lceil(4J_{n,p}/\varepsilon)^{1/p}\rceil$. Then the approximation error is at most $\varepsilon/4$, and $\|P_N\|_\infty\le2$. Write $P_N(x)=\sum_{k\in\{-N,\ldots,N\}^n}c_ke^{ik\cdot x}$. There are $D=(2N+1)^n$ coefficients, and normalized integration gives $|c_k|\le\|P_N\|_\infty\le2$.
Round their real and imaginary parts with mesh $\delta=\varepsilon/(8D)$, respecting $c_{-k}=\overline{c_k}$ so the rounded polynomial remains real. Each coefficient changes by at most $2\delta$, and hence the uniform change in the polynomial is at most $2D\delta=\varepsilon/4$. At most $(C/\delta)^{2D}$ choices are needed; allowing redundant choices only enlarges this bound. These rounded polynomials form an $\varepsilon$-net for the ball, and
$$
H(\varepsilon,B_{n,p})\le2D\log\left(\frac{CD}{\varepsilon}\right)
\le C_2\varepsilon^{-n/p}\log(1/\varepsilon).
$$
Here $D\le C_{n,p}\varepsilon^{-n/p}$, and the last logarithm absorbs the constant and $\log D$ for $\varepsilon<1/2$. Together the bounds give
$$
\boxed{C_1\varepsilon^{-n/p}\le H(\varepsilon,B_{n,p})
\le C_2\varepsilon^{-n/p}\log(1/\varepsilon)}.
$$
For the unit ball of $C(\mathbb T^n)$ alone, take infinitely many disjoint small neighbourhoods and a height-one continuous bump supported in each. Distinct such functions have uniform distance one. Thus no finite $\varepsilon$-net exists when $\varepsilon<1/2$, and
$$
\boxed{H(\varepsilon,B_{C})=\infty\quad(0<\varepsilon<1/2)}.
$$
The derivative bound was essential for <total boundedness>; mere continuity gives no common small-scale oscillation bound.
For the final composition estimate, interpret outer <periodic functions> on $\mathbb T^m$ by their $2\pi$-periodic extensions to $\mathbb R^m$. Since $p\ge1$, each $f\in B_{m,p}$ satisfies
$$
|f(x)-f(y)|\le\sum_{j=1}^m|x_j-y_j|
\le m\max_j|x_j-y_j|,
$$
by integrating its first partial derivatives along coordinate segments. This uniform <Lipschitz continuity> controls errors in the inner functions.
Choose an outer net of radius $\eta=\varepsilon/2$ for $B_{m,p}$ and an inner net of radius $\delta=\varepsilon/(2m)$ for $E$. For each $g=f(u_1,\ldots,u_m)$ choose corresponding approximants $f_0,v_1,\ldots,v_m$. No derivative bound for $f_0$ is needed: use the derivative bound of $f$ before replacing the outer function. Pointwise,
$$
\begin{aligned}
|f(u_1,\ldots,u_m)-f_0(v_1,\ldots,v_m)|
&\le|f(u_1,\ldots,u_m)-f(v_1,\ldots,v_m)|+\|f-f_0\|_\infty\\
&\le m\delta+\eta=\varepsilon.
\end{aligned}
$$
All composed centres are continuous. Multiplying the numbers of outer and inner choices gives the <entropy of Lipschitz compositions> inequality
$$
H(\varepsilon,F)\le H(\varepsilon/2,B_{m,p})
+mH(\varepsilon/(2m),E).
$$
Apply the proved upper bound with dimension $m$, and the assumed estimate for $E$. Fixed rescalings of $\varepsilon$ preserve its exponent and change $\log(1/\varepsilon)$ by only an additive constant. Therefore
$$
\boxed{H(\varepsilon,F)\le C_4\varepsilon^{-m/p}\log(1/\varepsilon)
\quad(0<\varepsilon<1/2)}.
$$
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