Can helicopters fly a loop?

Yes, some helicopters can indeed do loops, barrel rolls and even hammerheads … although it takes a specifically designed ‘rigid’ main rotor system. Such rotor systems are dominantly found on military-type helicopters with the exception of a few civilian helicopters.
Robinson R44  

For those who expect excellence and appreciate value, at last there's the R44 Raven. Quite simply, the Raven delivers the performance of an expensive turbine helicopter at an affordable piston price. Its sleek design, impressive speed, and unparalleled reliability give it the competitive edge in today's fast-paced, changing environment.

The R44 Raven is offered in two models:

The R44 Raven I offers turbine performance at an affordable piston price with high reliability, fuel economy, endurance, and comfortable hydraulic controls. This model is also available with fixed-utility floats as a Clipper I.

The high-performance R44 Raven II is designed for more payload, higher altitudes, and faster speeds with all the luxuries of the Raven I.