  As the fourth stage of the project multi-indexed orthogonal polynomials, we
present the multi-indexed Meixner and little $q$-Jacobi (Laguerre) polynomials
in the framework of `discrete quantum mechanics' with real shifts defined on
the semi-infinite lattice in one dimension. They are obtained, in a similar way
to the multi-indexed Laguerre and Jacobi polynomials reported earlier, from the
quantum mechanical systems corresponding to the original orthogonal polynomials
by multiple application of the discrete analogue of the Darboux transformations
or the Crum-Krein-Adler deletion of virtual state vectors. The virtual state
vectors are the solutions of the matrix Schr\"odinger equation on all the
lattice points having negative energies and infinite norm. This is in good
contrast to the ($q$-)Racah systems defined on a finite lattice, in which the
`virtual state' vectors satisfy the matrix Schr\"odinger equation except for
one of the two boundary points.
