Suppose we are given a directed system of spaces:
$$ ... \to A_n \xrightarrow{f_n} A_{n+1} \xrightarrow{f_{n+1}}A_{n+2} \to ...$$
such that for every $n$, there is some $m > n$ for which the composite $A_n \xrightarrow{f_{m-n}} A_m$ is null-homotopic (writing $f_{m-n}$ for the composite $f_{m-1}f_{m-2}...f_n$).
Is it then true that the space $A = \varinjlim_n A_n$ obtained as the direct limit of this system must be weakly contractible?
I am still trying to build intuition for how much information one can obtain on a direct limit from its constituent spaces, so I apologize if the question is a bit simple-minded.
I share Tyrone's doubts concerning the general case. Although I do know that much about Stiefel manifolds, I believe the corresponding direct limit has trivial homotopy groups. The Stiefel manifolds can be given the structure of CW-complexes, and I think the inclusions are cellular (you should check this).
Now, given a direct system of CW-complexes $A_n$ and cellular inclusions $f_n : A_n \to A_{n+1}$, the direct limit $A$ will also be a CW-complex. For any map $\phi : S^k \to A$ the image $\phi(S^k)$ is contained in a finite subcomplex of $A$ and therefore in some $A_n$. Hence $\phi = i_n \circ \phi_n$ for some map $\phi_n : S^k \to A_n$. But if the "null-homotopy-condition" is satisfied, we can conclude that $\phi$ must be inessential.