We perform a systematic DMRG investigation of the two natural spin-1 generalizations of the spin-1/2 Majumdar-Ghosh chain, the spin-1 J(1)-J(2) Heisenberg chain, where J(2) is a next-nearest-neighbor Heisenberg coupling, and the spin-1 J(1)-J(3) model, where J(3) refers to a three-site interaction defined by J(3)[(Si-1 . S-i)(S-i . Si+1) H.c.]. Although both models are rigorously equivalent to the Majumdar-Ghosh chain for spin 1/2, their physics appears to be quite different for spin 1. Indeed, when all couplings are antiferromagnetic, the spin-1 J(1)-J(2) model undergoes an effective decoupling into two next-nearest-neighbor (NNN) Haldane chains upon increasing J(2), while the J(1)-J(3) chain undergoes a spontaneous dimerization similar to the spin-1/2 Majumdar-Ghosh chain upon increasing J(3). By extending the phase diagram to all signs of the couplings, we show that both the dimerized and the NNN-Haldane phase are actually present in the J(1)-J(3) model, the former one adjacent to the Haldane one, the latter to the ferromagneric one, with an Ising transition between them. By contrast, the J(1)-J(2) chain only has a NNN-Haldane phase between the Haldane phase and the ferromagnetic phase for positive J(2). In both cases, our DMRG data are consistent with a continuous Kosterlitz-Thouless transition between the NNN-Haldane and the ferromagnetic phases.