Matrices possessing a low-rank property arise in numerous scientific applications. This property can be exploited to provide a substantial reduction of the complexity of their LU or LDL T factorization. Among the possible low-rank formats, the flat Block Low-Rank (BLR) format is easy to use but achieves superlinear complexity. Alternatively, the hierarchical formats achieve linear complexity at the price of a much more complex, hierarchical matrix representation. In this paper, we propose a new format based on multilevel BLR approximations: the matrix is recursively defined as a BLR matrix whose full-rank blocks are themselves represented by BLR matrices. We call this format multilevel BLR (MBLR). Contrarily to hierarchical matrices, the number of levels in the block hierarchy is fixed to a given constant; while this format can still be represented within the H formalism, we show that applying the H theory to it leads to very pessimistic complexity bounds. We therefore extend the theory to prove better bounds, and show that the MBLR format provides a simple way to finely control the desired complexity of dense factorizations. By striking a balance between the simplicity of the BLR format and the low complexity of the hierarchical ones, the MBLR format bridges the gap between flat and hierarchical low-rank matrix formats. The MBLR format is of particular relevance in the context of sparse direct solvers, for which it is able to trade off the optimal dense complexity of the hierarchical formats to benefit from the simplicity and flexibility of the BLR format while still achieving O(n) sparse complexity. We finally compare our MBLR format with the related BLR-H (or Lattice-H) format; our theoretical analysis shows that both formats achieve the same asymptotic complexity for a given top level block size.