  For any ordinal $\alpha > 0$, we show how to define a hyperexponential
$E_{\omega^{\alpha}}$ and a hyperlogarithm $L_{\omega^{\alpha}}$ on the class
$\mathbf{No}^{>, \succ}$ of positive infinitely large surreal numbers. Such
functions are archetypes of extremely fast and slowly growing functions at
infinity. We also show that the surreal numbers form a so-called hyperserial
field for our definition.
