Speaking to the asker:
From page 158 of the 2005 Crossfire Owner's Manual:
"DaimlerChrysler Corporation requires the use of 91 octane or higher premium fuel to minimize the potential for engine damage."
Spend the extra $4 per fill-up and get the 91 octane.
Speaking to the question itself:
Without specifics, and without defining "high" and "high compression" it isn't possible to determine an answer. However, all other variables being held constant, using a high octane fuel allows you to design a higher compression engine to take advantage of that octane number. The power then comes from the higher compression.
The compression ratio (CR) or induction type (turbo/super charged or normally aspirated) of the engine is immaterial unless you know the operating parameters of that particular motor! There are engines today that take 87 octane fuel at 13:1 compression, like Mazda's SKYACTIV, and some engines at 10:1 compression require 93 octane, such as the LS3 V8 in the Corvette. By the way, these are incredible compression numbers for unleaded gas, far higher than cars from the 1970's running 10:1 compression that used to require leaded fuel at up to 104 octane!
Different manufacturer's engines are designed around a specific octane number. Some can tolerate lower octane numbers for long periods by retarding engine timing or reducing turbocharger boost. This can be a designed operating parameter or a temporary measure to avoid engine damage. Depends on the manufacturer.
One owner's manual might specify premium fuel with no exceptions except in emergencies (Subaru WRX), while another owner's manual states lower octane merely reduces power with no long term harm (Nissan Maxima).
The only way to be sure is simply to follow the requirements of the manufacturer (who honors warranty work on the car) and engineers who designed and tested the engine.