  In this paper, we construct solutions of Lagrangian mean curvature flow which
exist and are embedded for all time, but form an infinite-time singularity and
converge to an immersed special Lagrangian as $t\to\infty$. In particular, the
flow decomposes the initial data into a union of special Lagrangians
intersecting at one point. This result shows that infinite-time singularities
can form in the Thomas--Yau `semi-stable' situation. A precise polynomial
blow-up rate of the second fundamental form is also shown.
  The infinite-time singularity formation is obtained by a perturbation of an
approximate family $N^{\varepsilon(t)}$ constructed by gluing in special
Lagrangian `Lawlor necks' of size $\varepsilon(t)$, where the dynamics of the
neck size $\varepsilon(t)$ are driven by the obstruction for the existence of
nearby special Lagrangians to $N^{\varepsilon(t)}$. This is inspired by the
work of Brendle and Kapouleas regarding ancient solutions of the Ricci flow.
