Homomorphisms from $\mathbb{R}$ to $\mathrm{Homeo}^+(\mathbb{R})$, or "fractional iterations"

This has some relation with this question, but it is obviously different.

In general, a homeomorphism of $\mathbb{R}$ which preserves the orientation may have or not fixed points.

If it has no fixed points, then it is conjugated to a translation and thus, one can easily construct such $\phi_f$.

The other case is not much more difficult, since one can consider the set of fixed points of $f$ (that is, such that $f(x)=x$) which is closed and then do the trick in the complement of that set and leave fixed the set of fixed points for every $t$ (when defining $\phi_f(t)$). Notice that in either case, there is in general no unique way to do this.

It is also interesting that the diffeomorphism case is quite different, in particular, one can easyly construct a diffeomorphism $f:\mathbb{R}\to \mathbb{R}$ such that there is no diffeomorphism $g$ such that $g\circ g =f$. This can be seen in the paper provided by Helge (in fact it has to do with distortion and the fact that if you take one contracting point, there are restrictions to construct, for example a square root, see Section 1 of this paper).

ADDED RELATED REFERENCE: In this paper, Palis gives a not so difficult proof that $C^1$-generic diffeomorphisms (which belong to a $G_\delta$-dense subset of $Diff^1(M)$) of a compact manifold, the diffeomorphisms are not the time one map of a flow.

I commented earlier, without finishing reading google's reply:

http://www.math.northwestern.edu/~wilkinso/papers/announce04152008.pdf

shows the answer is no if you replace $\mathbb{R}$ by a compact manifold. Key is googling for "Is every diffeomorphism a time one map?". I'm not sure what the answer for $\mathbb{R}$ is, and if it has been looked at.