The thing you actually liked
Search for games like A Dark Room and you get a genre label back: text-based, minimalist, incremental. None of that is the part that got you. Plenty of games are made of plain text and almost none of them do this.
Here is what the good ones share. In an ordinary incremental you can read the whole shape of the game in the first minute. There is a list of buildings, most of them greyed out, with prices you cannot afford yet, and the only unknown is how long it takes to afford them. In this handful of games the list itself is hidden. You do not know how many buttons there are. You do not know that a second panel exists, or that the resource you have been carefully hoarding is about to stop being the point. Progress is not a number going up, it is the screen growing controls you did not know it had.
That is the test I would apply to any recommendation here. A game belongs in this cluster if at least once you stopped playing for a second because the interface did something you did not know it could do. Games that only copy the look of these three, small text on a black background, are not the same game at all.
What this page holds back
The effect depends on not knowing, so this page works the same way our A Dark Room page does. For each game I describe the opening few minutes concretely, because you will see those anyway in the first thirty seconds, and then I stop describing events entirely. Nothing below names a twist, a reveal, or an ending. If you would rather go in completely blank, read the paragraph for whichever game you pick and skip the rest.
The three
A Dark Room is the one you already played, and it is still the best introduction to the idea because its first screen is the most extreme: two lines of text and one button. It is free in the browser, it takes most people two to four hours, and it does not have a prestige reset, so it ends properly instead of asking you to run it again. Made by Michael Townsend at Doublespeak Games, released in June 2013, source under MPL 2.0. Our full write-up is here.
Universal Paperclips is the closest match in structure and the furthest in tone. It opens with a button that makes one paperclip, a wire counter, and within a minute a price slider and a demand figure, so you are running a tiny business before you are running anything else. It takes the interface trick further than A Dark Room does, in that the panels it grows are not just new buttons but new kinds of decision, and it commits to finishing rather than looping. It is free in the browser at decisionproblem.com/paperclips, designed by Frank Lantz and programmed by Lantz with Bennett Foddy, released 9 October 2017 and built on Nick Bostrom's paperclip maximiser thought experiment. Paid mobile ports exist, which is worth knowing because the browser layout is a cramped wall of small buttons on a phone. One practical note for this site: unlike the other two, Universal Paperclips has no licence permitting anyone else to host it, so we link to Lantz's own copy and always will.
Candy Box 2 is the odd one and the one people bounce off. It begins with a candy counter ticking up on an otherwise empty page and two choices about what to do with candies, and it wants you to sit with that for a while before it gives you anything. What it becomes is much larger and much stranger than either of the others, an ASCII world you walk around with real combat and equipment underneath, and it is genuinely harder: it expects experimentation, it will let you waste resources, and some of it is deliberately obtuse in a way A Dark Room never is. Play it second or third, not first. Free at candybox2.github.io, by David "aniwey" L., released 24 October 2013 after the first Candy Box earlier that year, GPLv3.
Recommended alongside these, and only some of it belongs
Kittens Game by bloodrizer qualifies, and it is the best next step if you want a long one. It starts with a button that gathers catnip and it keeps opening new tabs across the top for hours. It is closer to a spreadsheet than to a story, so take it as the systems half of what you liked rather than the whole of it.
Spaceplan by Jake Hollands, published by Devolver Digital in May 2017, is short, finite, and paid rather than free. It is the most similar in feel to Universal Paperclips.
Cookie Clicker by Orteil turns up in every one of these threads and does not qualify. It is a very good game and its entire building list is legible from the first few minutes. That is the opposite of the property you came here for.
Where to start
Universal Paperclips if you finished A Dark Room this week and want the same shape again immediately. Kittens Game if you want something to keep open for a month. Candy Box 2 when you are willing to be confused for twenty minutes before it pays off, which it does.
This page is about the trick, games that hide their own shape from you. There is a second question people ask about the same cluster, which is which games could not be repainted into something else, where the story and the arithmetic are one object rather than two. That one is answered in games like Universal Paperclips, and it reaches a different list: A Dark Room only half qualifies there, and Candy Box 2 does not qualify at all.