Rapidly rotating compact objects could in principle possess the ergoregions (more general term than ergosphere) without event horizon. So it would be possible to extract energy via Penrose process from such an object. However, such object would also have ergoregion instability where it would be rapidly losing energy and angular momentum by spontaneously emitting gravitational and EM waves. Moreover, equations of state for real astrophysical objects such as neutron stars would not allow the formation of ergoregion without formation of a black hole. For technical details look for example one of the original works:
- Schutz, B. F., & Comins, N. (1978). On the existence of ergoregions in rotating stars. Monthly Notices of the Royal Astronomical Society, 182(1), 69-76.
and for some recent developments about ergoregion instability
- Pani, P., Cardoso, V., Cadoni, M., & Cavaglia, M. (2009). Ergoregion instability of black hole mimickers. arXiv:0901.0850.
Less rapidly rotating bodies, that do not have ergoregion would not allow the extraction of energy via Penrose process, as there would be no trajectories for massive point particles with negative energy.
Of course, rotational energy and angular momentum could be extracted from a non-black hole rotating bodies by a number of other means, possibly mediated through gravitational interaction, such as tidal friction.
Another interesting mechanism for such energy extraction is superradiance which is somewhat analogous to Penrose process only for waves rather than for particles. Rotating black holes display this phenomenon but also it could be observed in a rotating bodies without horizon (and also without ergoregion). A simple way to think about superradiance is that waves (EM or gravitational) around a rotating object can have modes with negative energies. If an infalling wave excites such a negative-energy mode and this mode is either absorbed by an event horizon (for a black hole) or dissipated by other means (for other rotating bodies) then the original wave would have a greater energy with the surplus coming from the rotational energy of a body. For a details see
- Richartz, M., & Saa, A. (2013). Superradiance without event horizons in general relativity. Physical Review D, 88(4), 044008, doi, arXiv:1306.3137.
or a book (with a free version at arXiv):
- Brito, R., Cardoso, V., & Pani, P. (2015). Superradiance. Lect. Notes Phys, 906(1), 18, link, arXiv:1501.06570.