  Kriz and Thomas showed that every (finite or infinite) graph of tree-width $k
\in \mathbb{N}$ admits a lean tree-decomposition of width $k$. We discuss a
number of counterexamples demonstrating the limits of possible generalisations
of their result to arbitrary infinite tree-width.
  In particular, we construct a locally finite, planar, connected graph that
has no lean tree-decomposition.
