Minimal number of cells of a CW complex (up to homotopy)  Given a finite connected CW complex $X$, one can ask what can said about the number of its cells. 
As an example, let's estimate the number of 1-cells: There is an epimorphism $\pi_1(X_1) \to \pi_1(X)$ from the 1-skeleton which is a connected graph and whose fundamental group is a free group on a subset of the 1-cells of $X$ [Hatcher, 1A.2]. Hence we have the lower bound
$$\text{number of 1-cells } \ge \text{ minimal number of generators of } \pi_1(X)$$ 
Conversely, given a presentation of $\pi_1(X)$ with a minimal number of generators $d$, there is a CW complex $X'$ with $d$ 1-cells and $\pi_1(X') = \pi_1(X)$ [Hatcher, 1.28].
Question 1: Can $X'$ be choosen to be (cellularly) homotopy equivalent to $X$ ?
Futhermore, by taking into account the cellular chain complex, it's not hard to see that the number of $n$-cells $(n \ge 0)$ is bounded below by the fact that we need (at least) 


*

*one $n$-cell for each direct summand of $H_n(X)$ 

*one $n$-cell for each direct summand of finite order of $H_{n-1}(X)$ 


As formula: 
$$\text{number of n-cells } \ge d(H_nX) + d(H_{n-1}(X)_{tor}) =: m_n(X)\qquad(\ast)$$
where $d(\cdot)$ denotes the minimal number of generators. 
It's known that if $X$ is simply connected, then $X$ is homotpoy equivalent to a complex $X'$ having exactly $m_n(X)$ cells in each dimension [Hatcher, 4C.1]. 
Question 2: Are there other classes of CW complexes where each $X$ is homotopy equivalent to a complex $X'$ that has $m_n(X)$ cells in each dimension ?
By the estimate above, a necessary condition for such a class is $d(\pi_1X)=d(\pi_1(X)_{ab})$, e.g. $\pi_1(X)$ solvable. 
Question 3: What's the best current bound for the minimal number of cells of (not necessarily simply-connected) finite CW complexes ? 
 A: To expand on my comment, there's a very general tool to manipulate CW-complexes, due to Whitehead.  It tells you when you can in effect remove a cell from a CW-decomposition via `elementary moves', usually called Whitehead Moves.   In smooth manifold theory there are parallel constructions -- people talk about "handle slides" and "handle cancellations".   This comes up in the proof of the h and s-cobordism theorems, which are the smooth-category analogue of the Whitehead moves.   Technically these moves have to do with a slightly more refined notion of homotopy-equivalence, called simple homotopy equivalence.  Provided the fundamental group of the CW-complex is trivial, simple homotopy-equivalences are in effect the same as homotopy-equivalances, but in general they're a little more fussy. 
What are the Whitehead moves?  On the 0-skeleton, it's the move where you collapse a maximal forest in the 1-skeleton. On the 1-skeleton these are moves where you can cancel a 1-cell using a 2-cell that's incident to it only once.  This is explained in detail in Marshall Cohen's "A course in simple-homotopy theory".  GTM 10 Springer-Verlag. 
