= Solution
For a based space $(X,x_0)$, the <homotopy group>
$$
\pi_k(X,x_0)=[(S^k,*),(X,x_0)]_*
$$
is the set of based homotopy classes, with its usual concatenation operation. A map $f:X\to Y$ is a <weak homotopy equivalence> when it induces a bijection on path components and an isomorphism
$$
f_*:\pi_k(X,x)\xrightarrow{\sim}\pi_k(Y,f(x))
$$
for every $k\geq1$ and every basepoint $x$. It is an <n-connected map> when it is bijective on $\pi_k$ for $k<n$ and surjective on $\pi_n$; equivalently, every <homotopy fiber> is $(n-1)$-connected.
A <CW complex> is built from a discrete set of zero-cells by successively attaching $k$-discs along maps from their boundary spheres, with the weak topology and closure-finiteness conditions. Its filtration by skeleta is the <CW filtration>.
For any space $X$, form its <singular simplicial set> $\operatorname{Sing}X$. Its <geometric realization of a simplicial set> is a <CW complex>, with one cell for each nondegenerate singular simplex, and evaluation gives
$$
\epsilon:|\operatorname{Sing}X|\longrightarrow X.
$$
The <Simplicial approximation theorem> identifies based maps and homotopies from finite simplicial spheres into $|\operatorname{Sing}X|$ with singular simplices in $X$. Consequently $\epsilon$ induces a bijection on components and isomorphisms on all homotopy groups. Thus every space admits a <CW approximation>.
The vanishing assumptions do not permit removal of all $n$-cells. Take $n=2$ and
$$
X=K(\mathbb Z/r,1),\qquad r>1.
$$
Then $\pi_2(X)=0$, and <homology of a finite cyclic group> gives $H_2(X;\mathbb Z)=0$. If a connected CW complex $Y$ had no two-cells, attaching cells of dimension at least three would not change the fundamental group of its one-skeleton. Hence $\pi_1(Y)$ would be a <free group>. A weak equivalence $Y\to X$ would instead give $\pi_1(Y)\cong\mathbb Z/r$, which is nontrivial and finite and therefore not free. No such $Y$ exists.
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