The exponential of an even multi-vector is related to rotation, but what is the exponential of a vector?
For instance, the exponential of a vector $\mathbf{v}=x\hat{\mathbf{x}}+y\hat{\mathbf{y}}+z\hat{\mathbf{z}}$ is:
$$ \exp (x\hat{\mathbf{x}}+y\hat{\mathbf{y}}+z\hat{\mathbf{z}} )=\cosh (\sqrt{x^2 + y^2+z^2})+\frac{x\hat{\mathbf{x}}+y\hat{\mathbf{y}}+z\hat{\mathbf{z}}}{\sqrt{x^2+y^2+z^2}}\sinh(\sqrt{x^2+y^2+z^2}) $$
What does it do geometrically?
To the best of my knowledge, the exponential of an odd-grade multivector can be interpreted as a hyperbolic rotation. The combination of regular (spherical) rotations and hyperbolic rotations gives you the Lorentz transformations, which are fundamental to relativistic physics.