There are so many variables when dealing with speakers, subwoofers, AVRs, and amplifiers. The RTiA9 speakers are particularly problematic when trying to make a single capability statement for every receiver, room, and crossover point. You can easily run the RTiA9 speakers with 30 - 50 watts. Now that I've said that, I have to caveat the statement with a dozen other qualifying comments. You will need to listen at lower volume levels, you will need to be crossed over at around 80 Hz, and you will need a quality dedicated subwoofer. PHEW, that's a lot of "needs".
Now let's talk about the average setup. RTiA9s are generally purchased due to their power handling capability and frequency response (even though they only dig down into the bass a little). If you run the RTiA9s at full range, with an AVR rated at 140W, with five channels driven with the AVR, and are watching an intense bass scene on a Blue-ray at near reference levels, you're going to clip the AVR amp and have massive distortion. Why, because the AVR will see dips in it's power output down into the 60W - 80W range depending on the scene, the reactive impedance of the speakers (the natural increases and decreases in speaker impedance through the frequency spectrum). I think we can all agree that 80W is not enough power to drive the RTiA9 in full range at reference volume levels. I don't know too many people that buy speakers with the intention of listening at low SPLs either.
The ideal setup for the RTiA9 would look something like this. First we have the RTiA9s crossed over at 80 Hz driven by a dedicated amp of around 200W - 300W. Even though the speakers can require massive amounts of power, they will generally only use around 30W for average movie watching and music listening. The more powerful amps are for those immediate peaks in power draw due to intense audio tracks. Headroom on the amp is key to a distortion free listening experience at high SPLs. Second we will need to have a dedicated subwoofer to take care of the low end business. Sure the RTiA9s work the low end pretty good, but they cannot produce clean SPL below around 40Hz. We will need to let a good quality powered subwoofer to manage the chest thumping, thus the 80Hz crossover for the A9s (and every other speaker too). This setup should provide hard hitting audio for even the most intensive movie sound tracks.
The budget setup isn't really a factor with A9s. If you have a budget that doesn't allow for a dedicated amp, I'd stay away from the A9s and go with A7s or bookies. You need a subwoofer for any setup to sound GOOD so the smaller speakers can do the same duties as the A9s with an 80Hz crossover. We all know that 75 percent or more of our listening is at -40 main volume and that is perfect for two A9s on an AVR with 100WPC or more. With that being said, the AVR neuters the A9s in the higher volume levels and you will inevitably be dissatisfied with the performance, always wanting to go higher and have more impact. The bottom line is that most AVRs will drive the A9s in moderate SPL at lower volume levels for most applications. We don't want moderate though. That's why I recommend a 200W - 300W dedicated amp for the A9s for any setup along with a good quality powered subwoofer. You'll be glad you made the jump to an amp and subwoofer early on. I am. Good luck and good listening.