Phonon lasers, exploiting coherent amplifications of phonons, have been a cornerstone for exploring quantum phononics, imaging nanomaterial structures, and realizing force sensors or phonon frequency combs. Single-mode phonon lasers, governed by dispersive optomechanical coupling, have been recently demonstrated via levitating a nanoparticle using an optical tweezer. Such levitated optomechanical (LOM) devices, with fundamental minimum of noises in high vacuum, can flexibly control large-mass objects with no internal discrete energy levels. However, it is still elusive to realize nonlinear multi-frequency phonon lasing with levitated microscale objects, dominated instead by dissipative LOM coupling due to much stronger optical scattering loss. Here, we report such a nonlinear phonon laser by employing a Yb3+-doped active LOM system. We observe a 3-order of magnitude enhancement for the fundamental-mode phonon lasing, compared with that in its passive counterparts. Above the lasing threshold, higher-order mechanical sidebands emerge spontaneously. Coherent correlations are also identified for the phonon modes. Our gain-assisted dissipative LOM platform no longer relies on any complicated external feedback control. Our work drives LOM into a new regime where it becomes promising to study mechanical properties or quantum entanglement of typical micro-size objects, such as atmospheric particulates and living cells, and build levitated force sensors with these objects for biomedical or astronomical applications.