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Phonon lasers, exploiting coherent amplifications of phonons, have been a cornerstone for exploring nonlinear phononics, imaging nanomaterial structures, and operating phononic devices. Very recently, by levitating a nanosphere in an optical tweezer, a single-mode phonon laser governed by dispersive optomechanical coupling has been demonstrated, assisted by alternating mechanical nonlinear cooling and linear heating. Such levitated optomechanical (LOM) devices, with minimal noises in high vacuum, can allow flexible control of large-mass objects without any internal discrete energy levels. However, untill now, it is still elusive to realize phonon lasing with levitated microscale objects, due to much stronger optical scattering losses. Here, by employing a Yb3+-doped active system, we report the first experiment on nonlinear multi-frequency phonon lasers with a micro-size sphere governed instead by dissipative LOM coupling. In this work, active gain plays a key role since not only 3-order enhancement can be achieved for the amplitude of the fundamental-mode phonon lasing, compared with the passive device, but also nonlinear mechanical harmonics can emerge spontaneously above the lasing threshold. Furthermore, for the first time, coherent correlations of phonons are observed for both the fundamental mode and its harmonics. Our work drives the field of LOM technology into a new regime where it becomes promising to engineer collective motional properties of typical micro-size objects, such as atmospheric particulates and living cells, for a wide range of applications in e.g., acoustic sensing, gravimetry, and inertial navigation.