how to calculate the fundamental group using intitutive ideas? I am trying to calculate the fundamental group of $\mathbb{R}^3- \{ x\text{-axis}\cup y\text{-axis}\cup z\text{-axis}\}$.
Idea: I think we can show it deformation retracts on 2-sphere minus 4 points.

As Qiaochu Yuan mentioned, $\mathbb{R}^3\setminus(\text{union of the $x$- $y$- and $z$-axes})$ deformation retracts onto the sphere $S^2$ with 6 missing points. To see this, note that $\mathbb{R}^3\setminus\{0\}$ deformation retracts onto $S^2$ by
$$ (\mathbb{R}^3\setminus\{0\})\times[0,1] \to \mathbb{R}^3\setminus\{0\} : (x,y,z)\mapsto t \frac{(x,y,z)}{||(x,y,z)||} + (1-t) (x,y,z) .$$
The same formula will retract $\mathbb{R}^3$ without the axes onto $S^2$ without the points $(\pm1,0,0)$, $(0,\pm1,0)$ and $(0,0,\pm1)$.
Now as pointed out by user61223, the sphere with 6 missing points is homeomorphic to the plane with 5 missing points, which has as fundamental group the free group on five generators. This is the fundamental group you're looking for.
