I find myself in the same boat... all those Steam and GoG sales, midweek madness and week-end special deals, and so on, games ending up at $5, $10, $30, it's often more than tempting and I do fall for my impulsiveness at times, and so I haven't bought a game in the past three months, but prior to that I bought nearly eight games in a period of a single month, all ranging in price from a mere $5 to $40. I haven't counted them all to update my "games list" but as of now I must own around 170+ games, and around 30+ of them were bought in the past six months or so.
Suffice it to say, when they're all installed, patched and modded to my tastes and ready to be played I just don't know what to do, which one(s) to play, and I often end up sticking with maybe four of them at most at any given time and play them for one or two months non-stop, then I switch with others and the cycle repeats, but I did buy some games that I have yet to play even if they are installed. So, in other words, yes it gets overwhelming. The best way to deal with that in my opinion, at least for me anyway, would be to install just the ones I know I want to play and finish, then uninstall them and install new ones and play them.
I wouldn't imagine having a collection of 300+ games, or even more... at that point it must simply become a collection habit and hobby, I wouldn't have enough of my life time to play so many games, and I have other priorities and obligations as well in life outside of gaming, but even if I dedicated my life to play games... nah, I wouldn't be able to.
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