A maxout is when a player reaches the game's score cap and can no longer earn points as a result. This is also known as a counter stop (カンスト kansuto).
Maxouts by game
The 1988 Sega arcade version of Tetris was the first official Tetris game to have a defined score cap, set at 999,999. By January 1989 the first maxout had been achieved on Easy difficulty by SPREAM-DIG.
In February 1989 SPREAM-DIG repeated the feat on Normal difficulty, and by March maxouts had been recorded on at least 10 different machines. Subsequently the focus switched to reaching the maximum score in the fewest lines, and by August, maxouts had been recorded within 600 lines on both Easy and Normal.
As with the Sega arcade version, the maximum score is 999,999.
It is not known when the first maxout occurred due to the difficulty of score tracking before the Internet, but given the calibre of arcade Tetris players it is hard to envision it not occurring in 1989 or 1990. A video exists dating from 2006, while J.O's high score page has reported a max out score going back at least to 2004.
The maximum score is 999,999. Because of the game's unique speed curve and leveling system, maxing out the Nintendo NES version requires not only survival skills but also line clearing efficiency, as it is nearly impossible to score significantly after reaching level 29.
Through use of a game modification device such as the Game Genie, the score cap can be raised above 1 million points; with the cap removed, scores of 1 million or higher will display a letter in place of the sixth digit (e.g., A00000, B00000, C00000, etc.). At time of writing, a few players have achieved scores up to letter C (1.2+ million). The current record for uncapped scores as of June 2020 is 1,390,780 points by Joseph Saelee; this was his fifth of seventeen times he scored a D. 
The PAL version has a different speed curve, reaching 1G at level 19 instead of level 29. However, since the framerate of the PAL version is only 50fps instead of 60fps for NTSC, it is relatively easier to survive at maximum speed than in the NTSC version. The first maxout on the PAL version was achieved by EricICX with a score of 1,096,300 points.
Unlike the NES version, Tetris & Dr. Mario does not cap the score at 999,999. Instead, the game rolls over the first digit, so that a score of 1,000,000 is displayed as A00000. The game continues in this manner until 16,777,216 (2^24), which is displayed as 0. However, the game will continue to record scores past 16,777,216 up to the four byte maximum of 4,294,967,295, though the displayed score will roll over every 16,777,216 points.
Although does not roll or cap when passing 1 million points, this event is still considered a milestone in the same vein as a maxout as it exceeds the maximum score that can be displayed with numerals alone.
Unlike the NES version, in this game Tetris & Dr. Mario the level score multiplier has a maximum of 10. For this reason, reaching 1 million points requires considerably more endurance than the NES version, but this is partially mitigated by the pre-gravity period making level 29-31 speeds more playable.
The first recorded (non-tool-assisted) game to pass 1 million was achieved by Koryan on 22 August 2019.
Other games with reachable score caps
Games with maximum score 99,999
Maxed out by Koryan.
Games with maximum score 999,999
Aside from Tetris Plus, and Tetris S, these games are considered relatively straightforward to maxout. Tetris Plus for arcade and consoles is somewhat harder, however the Game Boy version is seen as an easy game to maxout. The gameplay of Tetris S is almost identical to Tetris (Sega), and the game has the same maximum.
Games with maximum score 9,999,999
Maxout scores have been attested. Known maxouts include a game by Koryan in 2813 lines in 1998.
Achieved by Metallian in 2014.
Games with maximum score 99,999,999
According to an interview with one of the developers of Tetris (WonderSwan Color) a person left their PlayStation on for a year in order to reach a Tetris game's maximum score, but which game it was (Tetris X or The Tetris) is not specified.
Maxout scores have been attested.
Maxout scores have been attested in Endless mode.
Games with maximum score 100,000,000
The game displays the score as いちおくてん in hiragana when reached ingame, and as 100,000,000 on the high score table.
Games with maximum score 999,999,999
It is unknown if this score has been reached legitimately. A TAS exists that reaches the maximum score in one stage.
- Tetris (WonderSwan Color)
It is unlikely that this score has been reached legitimately due to the incredibly large number of lines that would be required (over 220,000 if no perfect clears are scored), but a TAS exists that takes advantage of perfect clear bonuses.
Games with maximum score 2,147,483,647
The game will not count the score event that takes it above 2^32-1, which means the game may stop at a slightly lower score. It is unlikely that this score has been reached legitimately due to the incredibly large number of lines that would be required (approximately 50,000 if only Tetrises are scored).
Games with score rollovers
Both the Atari arcade version and the Tengen NES version roll the score over after 999,999. For the Atari arcade version, since score is retained between credits it is straightforward to reach 999,999 by credit feeding. The Tengen NES version, on the other hand, rolls the score over to 100,000 after 999,999. Many players have been able to roll over the score in that version several times.
Tetris (Microsoft Entertainment Pack for Windows) uses a signed two-byte variable for score, and hence rolls over to -32,768 after reaching 32,767.
The arcade version of Tetris Plus rolls over at 1,000,000. This is in contrast to the console ports which stop at 999,999.