Many things admit log smooth morphisms to the standard log point. Consider a map \(\Spec k[P]\to \Spec k\) where \(P\) is a monoid and \(\Spec k\) is given the trivial log structure. Then
is a commutative diagram where the horizontal arrows are charts. The induced map \(\Spec k[P] \to \Spec k \times_{\Spec k}\Spec k[P]\) is an isomorphism and hence automatically smooth in the traditional sense. Thus the map \(\Spec k[P] \to \Spec k\) is log smooth.
This should mean that any fine log scheme \(X\) admits a log smooth map to the trivial log point, since the above diagram can be written down etale locally on \(X\).