For a connected pointed CW-complex $X$, let us write (as usual) $\Omega X$ for the space of based loops at $X$. I am looking for an example where the space $\Omega' X$ of all (unbased) loops in $X$ is not weakly equivalent to $X \times \Omega X$.



Example s.t. the unbased loop-space is not $\Omega X \times X$
Let $G$ be a discrete group. Let $P$ denote the groupoid whose objects are the elements of $G$, and whose morphisms from $a$ to $b$ are the elements $g\in G$ such that $gag^{-1}=b$. It is then a standard fact that $\Omega'BG=BP$. Less naturally but more concretely, if we choose a set $C$ of representatives for the conjugacy classes in $G$, we get an equivalence $\Omega'BG=\coprod\_{c\in C}BZ\_G(c)$. If $G$ is nonabelian then this will be different from $BG\times \Omega BG=BG\times G$. For example, you could take $G$ to be the free group on two generators, so $BG$ is a figure eight.


UPDATE: I can't immediately remember a convenient reference for the above "standard fact". However, if we regard $\mathbb{Z}$ and $G$ as groupoids with one object, then there is a functor $e\colon\mathbb{Z}\times P\to G$ given by $e((n,g)\colon a \to b)=g^n$. This gives a map 
$$ Be \colon S^1\times BP = B(\mathbb{Z}\times P) \to BG, $$
and adjointly a map $BP\to\text{Map}(S^1,BG)=\Omega'BG$, which is the one you want. Another way to analyse this is to consider $S^1$ as the pushout of two semicircles along a copy of $S^0$. After applying the functor $\text{Map}(-,BG)$ (and recalling that the semicircles are contractible) we get a homotopy pullback diagram 
$$ \begin{array}{ccc}
 \Omega'BG & \to & BG \\
 \downarrow && \downarrow \\
 BG & \to & BG\times BG
 \end{array}
$$
After thinking a bit about how the maps work, one can recover the previous description of $\Omega'BG$.


