Has been said already, but Arkham Horror LCG is excellent solo. Once you've built your decks or assembled one of the suggested ones, you take the role of one or two (one is haaard) investigators and navigate through a chapter of a Lovecraftian narrative, trying to collect clues to advance the story while doom tokens gather on the enemy agenda which can similarly advance. When either the story pile or enemy agenda pile advance, you flip the card and read the next part of the story before moving onto the next, which often rearranges the board and adds new enemies. Get a few candles and a two hour lovecraft music mix on the go and it's pretty atmospheric.
The core game revolves around each investigator taking a series of actions to play an item or upgrade into your inventory, move, investigate or fight the ghoulies. Most actions require a skill check which can be supplemented with cards from your hand to tip the odds in your favour, if you don't like doing skill checks, don't get this game. There are some complex-ish rules about monster aggro, when you're in and out of combat, sharing aggro between players, exhausting monsters etc. My common tactic is to bop the monster a couple of times with a guardian (fighter) class and then get one of the high agility investigators to engage it and dodge, which exhausts the monster and stops it from attacking on the monster's turn. There's an added layer where some monsters cannot be exhausted, or always aggro a specific investigator (e.g. person with the most/least health) and so on.
The objectives for each chapter change - sometimes it's discovering enough clues, sometimes it's defeating a boss, sometimes you're against the clock trying to do as much of something as possible before the game naturally ends in a push your luck scenario. Whether you win or lose, you note what happened and move onto the next chapter. Between chapters you can accumulate upgrades in the form of additional story cards in your deck plus upgrades and abilities you purchase with XP. Characters can also accumulate permanent damage which carries over to the next chapter.
The core box contains 4 starter investigators, a bunch of upgrades, 3 chapters which can be played through multiple times at multiple difficulties. However it's fair to say that you will eventually want more content after playing the initial chapters a few times.
From then on, each deluxe box (about 23 quid) functions as the start of a new cycle, which introduces more investigators, more cards, more monsters and 3 more chapters of a new campaign. You can mix and match all investigators and cards (everything is forwards and backwards compatible) giving you a reason to go back and play the old stuff with new characters and increasing your deckbuilding options. THEN, there's a cycle of six further chapters which are about a tenner each, which contains the next chapter in that campaign. So you're looking at 80 quid to complete a cycle, but a tenner a month to keep your game fresh isn't too steep if you play it multiple times. There's a nice element of collection too, especially if you buy/make yourself a nice box and some nice dividers.
The downsides are that the theme isn't for everyone - it's Lovecraft firstly, which might put off the wokest among us. The atmosphere is one of creeping dread and gathering doom (there is literally a doom counter) and you often feel like you're limping through a nightmare even when you're actually doing very well. It's a bit tough to teach people until you know it well yourself, and you need to buy two core sets in order to play 3-4 player.
If you're not sure it's for you and you want something a bit friendly, check out the upcoming Marvel Champions LCG, which seems like a very similar game with a Marvel theme and a few more concessions. Haven't played the LOTR one sorry.