We show that every henselian valued field $L$ of residue characteristic 0
admits a proper subfield $K$ which is dense in $L$. We present conditions under
which this can be taken such that $L|K$ is transcendental and $K$ is henselian.
These results are of interest for the investigation of integer parts of ordered
fields. We present examples of real closed fields which are larger than the
quotient fields of all their integer parts. Finally, we give rather simple
examples of ordered fields that do not admit any integer part and of valued
fields that do not admit any subring which is an additive complement of the
valuation ring.