The triangulated category of graded matrix factorizations for an exceptional unimodal singularity is known to have a tilting object by Kajiura-Saito-Takahashi and Lenzing-de la Pena. If we deform the singularity, then we lose the grading, which can be recovered by adding one more variable to the defining polynomial. The triangulated category of graded matrix factorizations of the resulting four-variable polynomial no longer has a tilting object, but has a classical generator, whose endomorphism algebra is the degree 2 trivial extension of the endomorphism algebra of the tilting object of the original category. In the talk, we will discuss the moduli space of A-infinity structures on this graded algebra, and its relation to 1. the positive part of the universal unfolding of the exceptional unimodal singularity, 2. the moduli space of K3 surfaces, and 3. homological mirror symmetry. If the time permits, we also discuss higher-dimensional generalizations and iterated singularity categories (i.e., singularity categories of singularity categories of ...) of non-isolated singularities. This is a joint work with Yanki Lekili.