What provides the greatest thrust in a high-bypass turbofan engine?
I analyzed a Pratt &amp; Whitney F100 turbofan last semester in my aerothermodynamics course, so allow me to answer this question. The short answer: the un-compressed air provides the majority of an engine's total thrust since the compressed air powers the engine.

Correction: I forgot to mention that the fans also compress entering air. That is, all air entering an engine is compressed a bit by the fan. Some of this compressed air enters the turbojet core and the rest of the fan-compressed air bypasses the engine core. I ignored this for the sake of simplicity, but I should have explained this since you directly asked about air compression in a turbofan.

Long answer: see below!


1. The fan(s)

Air enters the engine through a fan (or fans in the case of the F100 engine). These are the giant fans with a spinning insignia in the middle that you see inside an engine.

Update: The spinning insignia clearly show if the fan is spinning so workers don't get injured.

The fan(s) increases the pressure of the air that enters an engine. Some of this compressed air is diverted around the rest of the engine and is directed straight out of the engine. The bypass ratio is a measure of how much air bypasses the "jet" core (bypass air/core air).

2. The compressors

The rest of this air is then compressed through a combination of more fans and a converging duct. The data from my thermo project tells me that this stage increases the pressure of the core air by more than 10,000%, but I'm not too sure about that*. Sufficed to say, this core air now has a lot of energy--let's add some more :D

Quick note: The compressed air now has insignificant velocity relative to the bypass air. Most of the compressed air's energy is "in" its pressure (senior SE members, please correct me if I'm wrong).

3. The combustion chamber (aka, combustor aka magic chamber)

Now the core air enters the combustion chamber. Here the air enters small chambers, mixes with jet fuel, and is ignited. The main parts-of-an-engine diagram I posted makes it seam like the combustion chamber is one big part of a jet-engine, but really the combustor consists of a bunch of smaller chambers surrounded the main shaft of the engine. Here is a gif that shows what I mean:


How the combustion chamber operates is beyond my scope of understanding, but consider that a combustor is essentially trying to keep a candle alight in the middle of a hurricane. Awesome engineering goes into designing better and more efficient (hotter-burning) chambers.

4. The turbine (aka more magic section)

Now that hot and even more energetic air enters a the turbine section which consists of a diverging (increasing in area) duct and more fans. Whereas the compress "inserted" energy into the air, the turbines draw out energy from the air. As the air enters the larger (in volume) turbine area, it expands and spins the turbine fans which power the compressors and the fan. This Back Work Ratio (BWR) is a measure of how much turbine power it takes to spin the compressors. 

5. The nozzle

The still energetic core air is once again concentrated before being shot out of the back of the engine.This thrust, together with the thrust of the bypass air, propels the air forward following this model:

$F_{thrust} = \dot{m}_{bypass} \times \Delta v_{bypass} + \dot{m}_{core} \times \Delta v_{core}$

Where $\dot{m}_{bypass}$ is the mass flow-rate of air that is bypassed and $\Delta v_{bypass}$ is the change in velocity of that air as a result of the fan.

And $\dot{m}_{core}$ is the mass flow-rate of air that is combusted and $\Delta v_{core}$ is the change in velocity of the core air as a result of the fan, compressor, combustion chamber, and turbine.

The uncompressed air contributes about 60% of the total thrust. The "processed" air loses a significant portion of its energy to powering the engine. However, the compressed air still provides about 40% of the total thrust. Adding an afterburner can increase this contribution to 50%. How is this possible? Dead algae from a billion years ago.

The hydrocarbons in jet fuel pack a lot of energy into a small space and mass (two completely different concepts). Burning those hydrocarbons releases a lot of energy that powers the fan, compressors, and electric generators of an aircraft before pushing the airplane forward. This high energy in a small place/space is also why electric cars/anything weren't practical until LiPo batteries (story for another article). 

I applaud you for noticing that the uncompressed air contributes to an engine's thrust. I think the term "bypass" confuses some people into thinking that this air is "thrown away". it's not. The bypassed air is actually sped up by a series of fans and imparts forward momentum to the aircraft.

I'm studying Aerospace engineering right now (Whoop!), so my excitement at being able to answer this question rather side-tracked me from your original question, but I hope you enjoy the extra info on how a turbofan operates.

Additional info

The hotter a combustion chamber burns, the hotter the core air gets and the more energy it can provide for engine operation. Furthermore, hotter chambers waste less jet fuel and result in relatively cleaner and safer fumes. Increasing CC max temps and designing the rest of the engine (i.e., the turbine blades) to handle the increase in temp is the cutting edge of engine research. This is one of the most challenging and lucrative materials science/engineering problems in the world since efficiency is paramount in today and tomorrow's aviation industry.

Here is a nice article describing how GE, Rolls-Royce, and other engine companies sell thrust, not engines. It's a big long, so you may  read it on your next flight :)