3rd endless mode

The third endless mode is the third and final endless mode a player can reach before the random rooms glitch kicks in. It is preceded by the 2nd endless mode, and is reached by much fewer players (as little as a dozen known players, according to the Smash Hit page).
 * It is reachable at 28,000 distance but won’t have an ending (unlike the first and second endless modes), as at some point beyond the distance of 30,000, or stage 30 (score of 100,000 = stage 100), the Repeating Room Glitch will occur (2nd stage harder variant). The rare exception is that sometimes higher loops of endless mode (4th 5th etc) can be reached; but this appears to be an exception rather than a rule. (source needed that higher loops of endless mode usually do not happen; description of ‘2nd stage harder variant’ needed; more information on the exact distance at which the repeating rooms glitch occurs needs to be added).
 * If the Random Rooms Glitch is active when the repeating room glitch starts during 3rd endless mode, it will be re-doing the same (random) room (only one room) over and over again. In other words, the repeating room glitch takes precedence over the random room glitch, except that whatever room will be repeated will be a random one, not an actual room in endless mode that you would encounter if the random room glitch weren’t active.

It is possible to reach the 3rd endless mode without the Random Rooms Glitch. It is even possible to reach a stage high enough for the random rooms glitch to occur while using normal gameplay. It is easiest to reach third endless mode if you stash 1,100 balls or more by the start of first endless mode, which is usually accomplished by using the Ball Grinding Method, but can also be accomplished if a player attempts to conserve as many balls as possible, and gets very lucky (especially with the game's Room Generator). Since there is little margin for error in the latter strategy, a player should expect to have to do multiple runs, or, that not every run will result result in reaching 3rd endless mode. It is believed that reaching 3rd endless is easier in mayhem mode due to the boost you get in your ball count in the boss rooms.

Evidence
 Evidence of reaching third endless mode from normal gameplay: 
 * https://www.youtube.com/watch?v=0Hl-3M8XSDU (distance 35,551) (unplayable due to closed YouTube account, this was mentioned at one point)


 * MRK GAMES: distances of 32247 and 34558 have been reached, entering third endless mode using only standard gameplay. Videos: https://m.youtube.com/watch?v=YUuDV2KhTHw, https://www.youtube.com/watch?v=MRvMnhNDPEw&lc=UgzKXqFxpA_COdxlIIZ4AaABAg

 Evidence of reaching stage 30, i.e. the 'end' third endless mode, from normal gameplay: 
 * Benjamin Thoburn: third endless mode is again reached using standard gameplay. Video: https://m.youtube.com/watch?v=jmECbHYApPI
 * LorenzoDaGamer: Third Endless Mode at mayhem and classic mode using only standard gameplay. Videos: https://youtu.be/vB6uNLtBMOw https://youtu.be/eXe6VrrMXFY


 * Benjamin Thoburn: Stage 30 (repeating infinity/'checkpoint 15') has been reached from playing through the game normally. Final distance 35819. Video: https://www.youtube.com/watch?v=az0NOYmc7NA

GlitchLord

 * Reached a score of 35100 with the Random Rooms Glitch. See here: Checkpoint_5
 * The comment says:
 * "what??? checkpoint 13??? i have never got to checkpoint 13, there are checkpoints 0,1,2,3,4,5,6,7,8,9,10,11, infinity. If infinity is checkpoint 12, then there isn't a checkpoint 13. 5-5 is a fake level. My highscore is 35100 and if you use [the random rooms] glitch abuse you can get my score."

MondaysAreTheWorst (as of 26 January, 2020)

 * Rached 29310, beating the previous score of 28731 in Training Mode, it is the former world record of training mode.


 * Reached the score of 28258, beating 28080 in the Classic Mode.
 * Reached the score of 40543 in Mayhem Mode. It was previously 36000 (which was mentioned on a comment)

Tia'a Chon Ming

 * 32329 at mayhem, 25361 at normal, and 21039 in Training (comment here)