We study random simplicial complexes in the multi-parameter upper model. In this model simplices of various dimensions are taken randomly and independently, and our random simplicial complex Y is then taken to be the minimal simplicial complex containing this collection of simplices. We study the asymptotic behavior of the homology of Y as the number of vertices goes to ∞. We observe the following phenomenon asymptotically almost surely. The given probabilities with which the simplices are taken determine a range of dimensions ℓ ≤ k ≤ ℓ′ with ℓ′ ≤ 2ℓ + 1, outside of which the homology of Y vanishes. Within this range, the homology in the critical dimension ℓ is significantly the largest, and we specify the precise rate of growth of the ℓth Betti number. For the remaining Betti numbers in this range we give upper bounds that strongly decrease from dimension to dimension.