Such a subscription doesn't exist for Steam as there's too many 3rd party vendors on Steam for most of them to agree to allow access on a subscription based system.
HOWEVER, there are some subscription services available on some platforms.
* Humble Bundle's subscription service includes the "Humble Trove", where subscribers can download & install the DRM-free versions of the games within the trove. Subscribers can still play the games, even after your subscription expires (as long as you have the installers, as there's no DRM blocking you). Each month, you'll receive a bundle of games (which is a bit of a grab bag) that's valued OVER $100. The subscription runs $12 / mo (less for bulk subscriptions).
* EA's Origin has a subscription service called "Origin Access", which gives you access to a revolving library of games within Origin. HOWEVER, you lose access to those games once your subscription expires (until you renew your subscription). This subscription runs $5 / mo. or $30 / yr. for basic level & $15 / mo. or $100 / yr. for premier level.
* Microsoft has the XBox Game Pass for XBox One & PC (with games that support "XBox Play Anywhere") for $10 / mo. As long as you maintain your subscription, you'll retain access to the games on the service.
* Twitch Prime (as part of Amazon Prime) includes free games (& one channel subscription ) each month. Any games you pick up via Twitch Prime are yours to keep after your subscription expires, but you'll need to have the Twitch Desktop client to handle library of games you picked up.
Please be aware that no subscription service will have the complete library of a particular platform, so bear that in mind when selecting a subscription service.