I bet it's way less than you think - orders of magnitude.

What could happen versus what does happen are entirely different.

Doing some algebraic permutations computation here would be like claiming a 50,000 letter English document has 27^50,000 possibilities.

I mean no, there's words, and they only go in certain orders and there's all these rules.

Here's another approach: humans are pretty lousy when remembering large amounts of anything so let's say there's in practice, only been 100,000 unique games played over and over. Without the help of a computer or careful tabulation, I'm pretty sure no human would realize it because no human can remember 100,000 unique games.

Anyways, it's worth digging into the data to see what the variation really is. I bet the 90th percentile is embarrassingly small with a long tail that's far shorter then most think

edit: so I actually took 7.7 million games from https://www.ficsgames.org/download.html and did some basic processing on them. These are people ostensibly with ELOs over 2000 which is pretty decent just to see if I'll eat crow on this one.

Going in, I was expecting a uniqueness level to be something like 50-70%. Actual percentage of unique games over 7.7 million? 98.7%.

Alright fine.

Although I could try to do 1 billion games, I expected the distribution to be readily visible around 7 million.

Now as an artifact of the data, I made the games as compact as possible, potentially leading to ambiguity maybe. So a game might look like so

a3a5b3b6Bb2Bb7c4e6Nc3Nf6d4Be7Nf3O-Oe3c5d5exd5Nxd5d6Nxe7+Qxe7Bd3d5O-Odxc4Bxc4Nc6Qe2Rad8Rad1Ne4h3Nd6Ba6Bxa6Qxa6Qb7Qd3f6Qc3Rfe8Rd3Ne4Qc4+Kh8Rxd8Nxd8Rd1Qe7Qd5Nf7b4axb4axb4cxb4Qc6h6Rd7Qf8Qxb6Rd8Rxd8Qxd8Qxb4Qe8Bd4Kg8Qc4Nd6Qd5Kh8Nh4Kh7Qb3Qe4f3Qe6Qd3+f5e4g6exf5Nxf5Nxf5Qxf5Qxf5gxf5Bf2Kg6g4fxg4hxg4h5Kg2hxg4fxg4Ne5Kg3Nf7Kh4Ne5Be3Nxg41/2-1/2

Given this we can just run uniq with incrementing numbers and find out how things increase. I'm doing this on a pretty old laptop (3rd gen intel) so excuse me for cutting things a bit short

number of characters / unique entries / percentage duplicates

99 7470034 0.039416039621657628
98 7462496 0.040385363441781119
97 7454437 0.041421683508957363
96 7446241 0.042475620631500677
95 7437517 0.043597454142611958
94 7428583 0.044746291899176449
93 7419379 0.045929849399894973
92 7409325 0.047222709798876217
91 7399016 0.048548361067336399
90 7388091 0.049953224789125783
89 7376939 0.051387278814333581
88 7364995 0.052923177422393386
87 7352785 0.054493281408027117
86 7340118 0.056122151775432672
85 7326671 0.057851323625950024
84 7312421 0.059683754567414482
83 7297272 0.061631789397747494
82 7281670 0.063638076243272224
80 7247260 0.068062914748240111
79 7228770 0.07044057426456829
78 7209233 0.072952869233227302
77 7187947 0.075690070989017588
76 7166719 0.07841981442939705
75 7144539 0.081271977115830896
74 7119854 0.084446261873027284
73 7094951 0.087648579608837096
72 7068178 0.091091363720824936
71 7039566 0.094770627867995505
70 7009607 0.098623104961001351
69 6978335 0.10264442288391196
68 6944221 0.10703119826195528
67 6909408 0.11150785919986417
66 6871309 0.1164070722832925
65 6831848 0.12148142718723132
64 6790088 0.12685141428305979
63 6744878 0.13266504255419009
62 6698251 0.13866088518630681
61 6648086 0.14511168505848671
60 6594883 0.15195314634822232
59 6540926 0.15889156573829932
58 6481469 0.16653723917595897
57 6419070 0.17456122923325301
56 6355249 0.18276807660975847
55 6282195 0.19216221064468775
54 6209896 0.20145925798763076
53 6133584 0.21127234360201919
52 6048154 0.22225792783565479
51 5963420 0.23315401228435984
50 5870519 0.24510030469790289
49 5770633 0.25794480975187595
48 5668750 0.27104611232094422
47 5558836 0.28518013439112821
46 5443035 0.30007117547551587
45 5323607 0.31542861845637304
44 5192507 0.33228698311784588
43 5064129 0.34879532132158775
42 4925937 0.36656565792950735
41 4777302 0.38567887708631909
40 4626169 0.40511331817237839
39 4465105 0.42582480288508218
38 4301802 0.44682420429097458
37 4126391 0.46938059333470927
36 3948981 0.49219403707682896
35 3770310 0.5151696348833128
34 3581328 0.5394711411415466
33 3387960 0.5643366503548165
32 3202702 0.58815928132701434
31 3008299 0.61315788289287476
30 2810171 0.63863548833641626
29 2617450 0.66341779875536144
28 2413377 0.68965988152851743
26 2044415 0.73710531205655982
25 1849362 0.76218749819167997
24 1674682 0.78464988674290859
23 1499682 0.80715342462054207
22 1324053 0.82973784664289008
21 1167329 0.84989124361622848
20 1005747 0.87066933880105002
19 862832 0.88904701374837569
18 733254 0.90570966192613567
17 599187 0.9229495579983682
16 489782 0.93701812692123954
15 395141 0.94918816879710877
14 300280 0.9613865008348812
13 233128 0.97002168698093183
12 167915 0.97840753392729818
11 115118 0.98519678700915769
10 78474 0.98990889924908909
9 46588 0.9940091724420389
8 26581 0.99658190548385495
7 14316 0.99815908200996462
6 5390 0.99930689103336889
5 2659 0.99965807481590496
4 408 0.9999475346088339
3 180 0.99997685350389731
2 20 0.99999742816709969
1 9 0.9999988426751949

It would be interesting to see later, when I crunch larger numbers on a more capable machine, if these distributions generally hold. Of course it won't, it's not possible. But I'm wondering if it's greater than what the Shannon limit would predict.

An ancillary analysis would be to compare it to the possible legal permutations at a character count although this would of course require a board and rule model.

I would expect those percentages to decrease as the length increases and perhaps such a function can give more predictive heft to the actual "language" of chess in practice