  A strict bramble of a graph $G$ is a collection of pairwise-intersecting
connected subgraphs of $G.$ The order of a strict bramble ${\cal B}$ is the
minimum size of a set of vertices intersecting all sets of ${\cal B}.$ The
strict bramble number of $G,$ denoted by ${\sf sbn}(G),$ is the maximum order
of a strict bramble in $G.$ The strict bramble number of $G$ can be seen as a
way to extend the notion of acyclicity, departing from the fact that
(non-empty) acyclic graphs are exactly the graphs where every strict bramble
has order one. We initiate the study of this graph parameter by providing three
alternative definitions, each revealing different structural characteristics.
The first is a min-max theorem asserting that ${\sf sbn}(G)$ is equal to the
minimum $k$ for which $G$ is a minor of the lexicographic product of a tree and
a clique on $k$ vertices (also known as the lexicographic tree product number).
The second characterization is in terms of a new variant of a tree
decomposition called lenient tree decomposition. We prove that ${\sf sbn}(G)$
is equal to the minimum $k$ for which there exists a lenient tree decomposition
of $G$ of width at most $k.$ The third characterization is in terms of extremal
graphs. For this, we define, for each $k,$ the concept of a $k$-domino-tree and
we prove that every edge-maximal graph of strict bramble number at most $k$ is
a $k$-domino-tree. We also identify three graphs that constitute the
minor-obstruction set of the class of graphs with strict bramble number at most
two. We complete our results by proving that, given some $G$ and $k,$ deciding
whether ${\sf sbn}(G) \leq k$ is an ${\sf NP}$-complete problem.
