Three years, thousands of games, no improvement?

If I'm reading my chess.com statistics correctly, during the last 3 or so years I have played 3970 games. Pretty much during all this time have been stuck at about 1500 rapid rating and 1300 blitz rating, with no discernible improvement. (I haven't been playing a whole lot of rated games, but I don't see myself winning disproportionately many games against people at my level in unranked games either.)

Is this normal? Does there become a point where merely playing games will not help you improve at all?

well, i guess for some people. although, if you do not use and engine or have a teacher explain your mistakes, you will never improve, as you will make the same mistakes again. 

I have had a kind-of-similar experience. Three years ago I was new to the world of serious chess. I began playing online and in rated tournaments OTB. I was reading books, doing tactics, analyzing games move by with with the computer, etc. Then I got a little burnt out on the stress of tournaments and my online rapid rating was absolutely stuck at about 1250. I was helpless against anyone rated higher than that. So I decided to take a break from serious chess stuff and only played online blitz (5 min). That 'break' turned into a 3 year period. I played almost 9,000 blitz gagmes. Although my rating in Blitz did climb fairly steadily for a year and a half (from ca. 1,000 to 1,400), I have spent the last year and a half stuck there (wavering between 1300 and 1400). But I recently decided to get back into real study and tournament play. I've been reading books again, analyzing games again, reviewing openings, endgame studies, doing practice and tactics again, and playing rapid games online (30 min) and analyzing those in detail. In a little over a month I've increased my online rapid rating to 1470 (a few days ago at least). 

My summary is: If you're just playing casual games for fun, you are not improving much and definitely not quickly. Sure, I learned lots and lots of patterns by heart in the 9,000 blitz games I played in 3 years. And my tactical eye got better. But improvement truly requires detailed, deliberate STUDY. When I have heard people debating whether or not chess should be considered a sport or a game, I always say: it is not truly either; it is primarily a subject of study, like an academic subject, albeit a very niche 'academic' subject. (It is also a game.)

1 500 rapid is a reasonable rating. What have you been doing except playing games? If you were just playing the entire time, it is probably normal to stay at that rating, or improve slightly. 

Have you been trying to study chess in some way (books, chessable courses, videos), have you been practicing tactical puzzles, studying master games, analyzing your own games apart from playing?

1500 is reasonable, and many people (like myself), never get a solid rating above 1500, although I can win against 1600 and very occasionally against someone rated 1700, but I have USCF and rating from other chess servers and never have been able to have a solid rating 1500 or above.

Probably the best USCF game I ever had was a draw against an 1800.. and the opponent was pretty pissed I pulled it off. At least since college, my USCF rating has been between 1425 and 1475.

On that note, the 1500 rapid rating is here, at chess.com. I have absolutely no idea what it corresponds to in FIDE Elo rating (it's my understanding that these two rating system can vary by quite a margin). Would love to know, but unfortunately where I live there aren't really any options to play live games or any games where your FIDE Elo could be measured.

(Not that I would be extraordinarily eager to play tournaments. I have played in many tournaments a long time ago, and I don't really like them. Too stressful. Casual games are much more enjoyable.)

Ah, Grasshopper.

I learned this. I didn't learn it from any chess players, directly or indirectly. I learned it from a martial artist. Hist style was 8 Animals Kung Fu, taught father to son, down the generations. It's origin is 5 Animals Kung Fu from the Shaolin Temple (for real). Their family added 3 more animals.

The Chinese are so smart. They can use different styles to fight. You don't advance because you are doing the same thing over and over again. Learn a different style of fighting, a different opening.

"kărätê" has 3 steps of learning.

Kata. Form. Reading books.

Randori. Sparring. Sparring against computers.

Kumite. (Everybody has seen Bloodsport). Real fighting. Play real games, at least 40 minutes each.

There you go, Grasshopper.

I actually learned that in the video game Budokan for DOS.

The best resource I know of is this:

https://chessgoals.com/rating-comparison/

Bear in mind that this is an approximation and individual cases might be different.

In the coin operated video game Karate Champ (for 1 player), it demonstrates this concept.

You go to the "dojo" and practice your kata.

Then you spar with the other students in the dojo.

Then you go to the kumite.

I reviewed one of your game to see which areas you may need improvements.

2. Lack of opportunity to seize " e" file in opening

3. Middle game has no plan ( king attack, control centre or queen side push)

Playing isn't how one improves.

Studying is how one improves.

Playing is just to practice what you've studied.

Playing also tells you where you need to study.

Playing alone without review may make you doing the same mistake again and again.

Whether you play 3 mins blitz or 30 mins rapid, reviewing the game is always the key.

Study of 

1.opening theory 

2.endgame drills

3. Tactics

Studying for improvement is tedious, boring and hardwork. No one like it. But do it if you are serious in the game.

We're not in total disagreement. I consider reviewing part of "studying".

Though, studying also extends past reviewing.

There are lot of ideas in chess that have been found by strong grandmasters--significant ideas--that reviewing alone won't help one learn.

But yes, reviewing is definitely helpful.

It's true, also, that not everyone enjoys studying. Some love it, though. A lot depends on how it's done, and what kind of player one is striving to be.

I partially disagree with this. Playing is a necessary component to improvement, especially at lower levels. You may improve some by just studying, without playing, but nowhere near as much as you'd hope. Below 1800 level, I'd say actual play time, at slow time controls, should make up roughly 50% of your chess time if you're serious about improvement.

What would be the best and most convenient (and preferably cheapest, if not free) way of studying chess theory at this level?

Chess is no different than most fields of human endeavour where the population distributes according to a bell curve: the more you want to move to the right of the median, the harder it gets, generally speaking.

For opening,

1. Use of free opening explorers, e.g https://www.365chess.com/

Study mainlines. If you are in doubt why they move those mainlines, use engines why those moves are refuted. Specialize in a few openings only and apply your mainlines, and get advantage in opening if opponents deviate from mainlines. 

2. Review of hundreds of master games to get idea of your opening lines (e.g Sicilian ) how they proceed their plan in middle games. Some CD has millions of games. Games are also freely accessible in chess.com and other internet rescources

3. Tactics

4. Endgame drills, you can use your own game positions , and play against engines in the positions where you misplayed endgames. Endgame knowledge are critically improtant to decide where to keep your game pace, keep in middle game vs simplification into endgame. (certain pawn advanced positions or extra pawns are winning in endgame but some are not, e.g rook endgame with extrapawn in same side are usually draw)