Well, there are free alternatives to avast!: AVG, Avira, BitDefender, etc., and it has no pop-ups of any sort: It is MSE.
Having said that, you might want to have a look at the diminishing support it is now getting from sites that once recommended it, including Microsoft: http://www.howtogeek.com/173291/goodbye-microsoft-security-essentials-microsoft-now-recommends-you-use-a-third-party-antivirus/ One of the reasons for support over the years was because of no nag pop-ups, and also the least impact on system performance and resources of all a/v's tested.
For your specific needs, MSE may be exactly what you need. Just be aware, unlike avast!, its' antivirus engine can be disabled/corrupted when malware enters and attacks your system, and then, when that happens, one has no real-time protection whatsoever following this sort of incident.
It now is a risk one has to take when one runs MSE as their a/v of choice.
Actually, we're all in the same boat here: All free versions of antivirus programs except MSE have adverts and pop-up windows of one sort or another; some are more intrusive than others, so it is then a case of pick your lesser evil here.
None of us as users create or maintain these a/v softwares, so essentially, what you desire, is in a category of a wish list, as one really cannot change any settings to what they want if that setting is not there. If we had control of design and implementation, that would be another story....
WYSIWYG.
So what you are saying is that all 3:rd party a/v:s have ad-popups (not just warning/update/usefull ones)? In that case, I am more assured that my decision (taken yesterday, I was still thinking of it when I posted this thread) to uninstall avast and go for windows defender was the right one. I simply cannot accept popup advertisement pollute my computer environment.
Imagine all you installed apps/programs taking the same right. Your media player pop-up-offers you music or movies perhaps based on your media habits, your browser advertises pages you might want to visit based on traffic, Acrobat Reader sees that you are often working with economy/business and thinks you might be interested in books about that, your cloud app pop-ups offers about a larger subscription, Windows/Ios itself takes the right to offer you Microsoft or Apple applications you might be interested in, your Nvidia/ATI app asks you every day if you are REALLY sure you don't want to upgrade to a faster video card, your webcam is surely not the most recent product and maybe you want to "upgrade" it, your sound card, your hard drives, your bloody USB-memory...
You see where I'm going, it would be a lot of pop-ups trying to get your attention. Then the competition forces them to take the next step in order to be seen. And what is this next? Changing the background to "Offer of the day"? Playing sound-ads through the speakers? Mailing my e-mail contacts? Uploading my backup-contacts and text them? Pausing your on-going media to play/display offers? Changing your boot-screen? Installing "free trial"-apps on your computer?
No, a line has to be drawn and everyday pop-ups are way crossed that line. Android users drew the line when fully "legit" apps were bundled with advertisement apps that once in a while activated you a notification event about a generous offer (sounding like you got a text/e-mail/tweet/etc and displaying a message in the notification bar on top). No matter how far advertisers pushes the line (as in increasing user tolerance), it is always motivated by "it's free, we need ads to keep it free, you haven't payed so don't complain, no one forces you to use the product, etc, etc". The fact is that often none of that is true, the free user base is just another product model. Browsers, as an example, makes money from in-browser searches and sometimes licensing.
Sum summarum: Even though am primarily motivated by the fact that pop-ups are plainly annoying, I also have a wider perhaps more rational reason. I get that everyone isn't there yet, as in not enough annoyed by the pop-ups, but I personally am and that is why I have chosen to uninstall avast and post this thread.