I've got three games to recommend - they're fairly well-known, but I thought I'd put them out anyways, just in case someone entirely new to visual novels stumbles across this thread.
The first one - which isn't technically free - is Fate/Stay Night. Fate Stay Night is a Japanese game which follows a young man named Shirou, who due to happenstance is dragged into a secret Underworld of mages. Sounds a bit like Harry Potter, except these mages also happen to be currently involved in a battle royale for the ultimate wish-granting device ever created - the Holy Grail. To cut a long story short, the Holy Grail is summoned using the spirits of seven dead heroes, and accessing its powers requires the sacrifice of all seven spirits, each of which are summoned by seven different mages. Problem is, the Grail's only granting one wish for all seven people. Instead of talking it out like reasonable individuals, the obvious solution that the mages take is to start the Holy Grail War, using their summoned spirits (servants) to kill the servants summoned by others.
If you ever wondered whether Hercules would beat Cu Chullain in a fight, or whether legendary Japanese Samurai could take on European knights - this is your kind of visual novel. The servant vs servant battles are pretty glorious, while in the background the mages try to figure out who opposing servants are to exploit their weak spots.
And Shirou just happened to get caught up in this war, because he summoned a servant himself. Saber, a young woman wielding a sword who is absolutely determined to win the War. Not going to spoil too much, but... Fate/ Stay Night constantly surprised me when I first played it, and you never really get a full picture of what was going on until you've played through every single route of the game. Each route kind of reveals what seemed to be whimsical decisions from other mages or servants at different points in the game, and while I won't say it dispenses life changing wisdom or anything, it's at least not too dumb for an otome game.
Now, going back to how it's not technically free. It was a professional game released by a pretty big franchise in Japan. That said, it's never been released outside of Japan officially, nor has it been officially translated. Dozens of fansites have English translations up and running, and unless you're fluent in Japanese or tech-literate enough to mod an English translation into your Japanese copy of the game, chances are you'll have to get the game off fansites. There's also a problem in that the best graphics and animations were reserved for the most recent PS Vita release, and the only way you can get that version on a PC (as opposed to a PS Vita) is by downloading a modded version off English fansites. Or Japanese fansites. Getting your hands on a legitimate copy is not impossible - but it's pretty difficult.
The second game which I'd like to recommend is much easier to access, and also completely free. It's Narcissu. It's another relatively old game, but was released in Linux, iOS, Windows, and even Android. The story follows a young man whose been diagnosed with a debilitating disease. His family, always more focused on work than on their only child, effectively gives up on him and sends him away to a hospital ward where they expect he will soon die.
What his family did not expect was that he would go full-on grand-theft-auto with his fellow walking corpse, steal his fathers' prized car, and go on a road trip.
The story is quite short - less than an hour long - and it gets a little corny at times, but I found it highly enjoyable. There aren't any choices you can make, but the narrative was surprisingly mature (and I don't mean in a sexual way. It's just the way it handles imminent death hanging over everyone's heads all the time).
The third game I'd recommend is also technically not free - for much of the same reasons, but also possibly because no reputable English gaming or publishing company would even consider licensing it. Saya No Uta, better translated (IMO) as Song of Saya, is a short, but still rather disturbing visual novel that launched its writer into infamy amongst both western and Japanese V/N circles.
The story begins as we are introduced to Fuminori, a young man who lost his family in a tragic car accident that left him with seemingly slight brain trauma. What only he knows, however, is that the accident appears to have triggered some form of severe agnosia in him. He sees, hears, touches, smells, and even tastes the world very differently from everyone else. Former friends are transformed into sacks of writhing meat, voices of concern gibbering screeches that he can barely understand, and his clothes, his food, and even his house looks, feels, and smells to him as though they were the innards of some decaying monster. He keeps this a secret for a very reasonable fear of being locked up in an asylum. His frail grip on what remains of his sanity, however, is reinforced by a single pure thing in his warped world. A young homeless girl named Saya, who looks, acts, and speaks just like a normal pre-adolescent girl would have.
If you think the scenario already sounds messed up, the story gets a lot worse. Expect rape, paedophilia, kidnapping and torture, murder, induced insanity, cannibalism, gore (kind of duh to be honest, but I'll put it in anyways), and literally the annihilation of the human race in gory detail.
Funny thing about the writer, by the way. Upon being interviewed on this visual novel, he reportedly said something along the lines of "I'm really happy because this is the first time I could write a genuine happy ending." Said happy ending literally annihilated humanity. Not even kidding. was not happy at all. One issue with the game was that there was in fact no happy ending, or even a happy journey to a sad ending for that matter.
So, this was a really long post. Longer than I expected it to be. But I hope you like one, if not all, of these games.