I have thoughts!
You said “board game”, so I’m not sure where cards fit in on things, but…
the Pathfinder Adventure Card Games are pretty terrific. You can play the first adventure set on the Steam/iOS app. But be aware: there’s a bug – that will never be fixed, apparently – on the Steam version that requires you to go through some extraordinary measures to finish the full campaign.
The physical boxed versions are very good though, if a bit pricey as you add each card set.
I will echo the sentiment for Robinson Crusoe. Just be sure you get the Portal Games version, as it has rules that are actually readable. What’s weird with Crusoe is that the first expansion – which was for the Z-Man Games version, but is perfectly compatible with the later edition – is pretty terrific, and offers the campaign play/persistent setting you’re looking for, and the rules mostly make sense. The second, big box Crusoe expansion is filled with beautiful components and great ideas and a theme that looks and feels amazing…and an absolute farrago of poorly written rules that are every bit the issue that the original rules to the game were. Getting through even a first game here has been a slog as a result.
Arkham Horror LCG is also all cards – but plays very much like a boardgame. Just know going in that if you get hooked on the game, you’re going to be buying regular installments of cards for new campaigns and scenarios. With that being said, I am now willing to take up the unpopular opinion that this game is VERY replayable. I’ve played through the first two campaigns (the one in the core box, and the Dunwich campaign that follows) at least a half-dozen times, trying different investigators, deck builds, etc. And I’ve enjoyed the replays a great deal. Though I know what’s coming in the story beats, there’s enough variety in the way the game plays with different investigators/builds and campaign scenario branching paths that none of my campaigns have played out anywhere close to the same way.
Nemo’s War is amazing. Get it while you still can.
Now let me champion an underdog of a solo game that stupid Tom got me hooked on: Elder Sign. Let’s be clear on this: it is absolutely, positively Cthulhu Yahtzee. And the thing is, for as simple as the core box physical game is, the computer versions are even more simplified.
But here’s the thing on Elder Sign: the yahtzee mechanic is kind of fun, but the core game feels really sort of empty. It’s not well-married to its theme. But then with Elder Sign’s expansions, something kind of crazy happens. So, the big criticism of Fantasy Flight is that they expansions their games to death, almost literally. Arkham Horror became exasperatingly unplayable with all the expansions tacked onto it. Ditto for Eldritch Horror, which is a GREAT solo game if you stop shortly after the first small-box expansion.
But with Elder Sign, the rules are so minimal anyway at the start, and the game so kind of lacking in narrative and theme that it ends up the rare exception: it’s the Fantasy Flight game that gets better with the expansions. For instance, the second expansion, Unseen Forces, really dresses up the basic game play a bit with more cards, investigators, and a cursed/blessed mechanic. Then Gates of Arkham comes along and REALLY kicks things ahead with a new Adventure system that gets you out of the museum, and a bunch of new tactical and strategic choices in cards that help to add narrative and theme…and also increase the number of interesting decisions you have to make in the game. And then you get the three “narrative” expansions, the “Omens of…” boxes that are set in arctic Alaska, the oceans deep, and Egypt. These are narrative heavy, campaign-feeling expansions that offer the tightest feeling of the Arkham theme.