With the right mix of materials, moisture, air and mixing, you could get compost in about 3 or 4 weeks. With the wrong mix, it will take longer. How much longer depends on too many variables to venture a guess.

If you're in a hurry, you could try mixing it up a bit to see if it speeds the process. If it's dry you can add some water. If it doesn't have enough N you can add some grass clippings or coffee grounds.

If you're not in a hurry, you can just wait and use it when it's ready.

I hear many people say you can get compost in a matter of weeks but that compost is not properly aged and could still have compounds in it that will be hard on some sensative plants.

Anyway, how long compost takes will depend on what you put in the pile, how active your mix gets, how much you need to let in age depending on ingredients and type of plants it is to be used for, and how perfect you expect your compost to be.

There are different schools of thought on compost. Some people work hard to get compost in three weeks. Others will continous compost where it will take two years. Continous compost will usually fill a pile for a year or untill it is full without turing and then let it age for a year before use.

If you are composting the "no-nos" (meat, dairy, pet/human manure etc) then you need to have a very active pile that is allowed a good long curing time before use. If you are mainly composting grass clippings, leaves, and kitchen scraps, you can probably get away with using the compost as soon as it looks/smells acceptable to you.

Congrats, it sounds like you have a good thing going, keep it up.

Thanks for the comments! I added some more grass to the pile today, and I asked Starbucks to hold onto its grounds for me. I am going to get them early evening and was thinking about mixing them right in. Are there any rights/wrongs associated with adding coffee grounds?

The coffee grounds from Starbucks are often very finely ground. This depends largely on whether the grounds are mostly the espresso "pucks" or the grounds from the drip coffee. The pucks can get pretty compressed and repel water if they're layered too thickly (even if you break them up) so it's a good idea to try to mix the grounds in with the rest of the material you're composting.

How long from start to finished compost depends on many things, the right C:N ratio, just enough and not too much moisture, and turning at just the right time. You could have good compost in as little as 14 days (link below) but that is a lot of work and you cannot compost diseased plant tissue in this since there is simply not enough time for immunities to develop.
There are those that get finished compost in 3 or 4 months by closely monitoring the process and turning the pile at the right time so that the material on the outside gets into the center where the bacteria that digest that material are most active. This too is a fair amount of work.
Many of us use a more lassiez faire approach and allow our compost to work as it will and most usually get finished compost in about 1 year. In your 3 x 3 x 3 pile those PVC pipes for aeration are unnecessary and may cause a lower temperature than some of the bacteria that digest your material find they can work in slowing down the process. Compost piles smaller than 6 x 6 x 6 do not need aeration aids, unless you soak the material that goes into that pile so air infiltration will not take place.

Here is a link that might be useful: Compost in 14 Days

I don't know whether it's true that most of us get finished compost in a year. I don't have enough space to have that kind of compost.

I don't get compost every 14 days, but I usually get compost in 3-6 weeks. I don't find it all that labor intensive. I've got a bin that has a door near the bottom. I periodically take a shovelful from the bottom and put it on top. If it doesn't settle on its own by the time it reaches the top of the bin, I use a bulb auger to mix it some more.

There are those who worry that if you don't age your compost for many months, you'll suffer dire consequences. I'm not one of them. I usually spread compost on the lawn, especially around a tree that has problems, and only spread compost on the garden in the fall, so maybe that addresses the concerns of those how feel all compost needs to be aged.

From what I've read, if the compost heats to at least 135 F for two to three days, the pathogens are destroyed. The reason I don't spread the compost on the garden except in the fall is that's the only time the garden's completely uncovered.

@kimmsr

"You could have good compost in as little as 14 days but that is a lot of work and you cannot compost diseased plant tissue in this since there is simply not enough time for immunities to develop."

Do immunities continue to develop during the curing/aging process and what sort of time period is required?

Very possibly the soil bacteria too can develop the immunities to disease pathogens since there is also evidence that turning diseased plant tissue into your soil, along with good quality compost, can help eventually. For compost there was some research many years ago that indicated the 14 day compost did not develop the immunities that were present in compost that took longer. Since this was before the internet these studies most likely were printed in Organic Gardening Magazine, Mother Earth News, Countryside Magazine and you would need to find those to see the research.

I know a company which is pretty good at soil diagnostics (bodemverbetering). It is a dutch company, though they can probably give you some advice on this.