Large numbers

In one of the homeworks, you had to estimate how large eee10 is. We can compute with the calculator:
e10 ~ 22026  
ee10 ~ 109566
eee10 ≤ 10109566 and because 10x ≤ ex/3 one also has 10ee10 ≥ e109565
So one can say that the number has between 109565 and 109566 digits. One could not even write down a number with 1090 digits because there are less elementary particles in the universe than that. The Eddington number is the number of protons in the observable universe. It is believed to be less than 1080. These numbers are astronomical, in the proper sense of the word. It is interesting that Archimedes (whose 2300th birthday we celebrate) has estimated the number of sand grains in the universe. The Sand reconer got about this number as an upper bound. He estimated that 8 * 1063 grains of sand would be needed to fill the universe. source.

The graph of eee10

We can plot the graph from 0 to 1, but already from 0 to 2, Mathematica refuses to plot it. The online Mathematica version is more forgiving. Here is the graph in Wolfram Alpha.

The integer digits of ee10

This can still be written down by typing "Floor[Exp[Exp[10]]]".

    9387512218583594745539690918995541531009584046561616020605905299722770917
56643174045789642161542666221186248885109035770936484011458213525445995968952
70247574596016987708892173045619561154773552332646127879507727370720244156666
89717132346997543823972950588115027099471233278301685635826041260906156684599
60118464370026156340540512782031536399864955827898930228192861200154758465191
02210580121296340785188272177546527789671458315913063409077962986525224155263
39936017406786560163470257240527152343030154717865370249654584485730325693051
28310442307969393767496402833906860782276498013369408833552780496177418704871
61090641206302820519934880060121043197303921265539759574717306841145256598423
82124631597947361058317857823136048024565564879156799146524260810882789386856
21057029735207835371714634024819324045619273082891703376077311407879899701527
71056536678462271174752222106153506963082447757381847206338882726585357901261
14630606718658030099138667482000431743000310258510607880111132902137759884550
03444104374587152446664754034185245502722610986693316126018713036076457984234
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44764647764935174299447497230517999301695766692572276757314211830689544507845
53074932288016785405953040627509878429870427125048126461777687635836397376421
55615118494445470122034294488138163368425603721032870655735691865832849471396
94822627835890272527937294949499675090500021125428922184933767750889593157843
50764563443954000659796445675038276022040397487838714696775097192612931481222
69840538126319378046600534250679279176763095587723381922073582563604820575182
94684211680738172460683761506331350883866669584806866251308646786716391366126
57539240481455523034319070506067615080081116788976722010641327901261346292680
24942049098518123523586515565068093060950323410029595422000887605645825348269
25855094380906348272805601638280597428349084596784891569257092003945756043000
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80386552198013164550041661592320218811008880513046066426467553053008932316676
85330500751696952618098893488648558425330451264807387444926521878298132305319
11702891112015046495053208408158558469437351867712204612151200656558072752314
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38458773074323880277439150830092967167733363638700664389528705475479887450424
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63188972680992742999503903632402048665111683710404368046515868845924246524730
74950951045415967220772593640701150659214509940578045731452239313271417385909
62926788496608174175736430868226000600749262873018823117663375089720904075318
40953135342702907880038298976547439818571671687233599775010429428532013787290
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15658086284300499089036451999367871991158574307085061655035039103960235235918
25158961330605582416596987903788962809051394922872032948110927498843142646188
59955738303774121794327094081817918296662592266555863806618151815839598868931
36221824248514605056757561707436371122093258833418640032212044776752513867270
18632135400615961805386849287351258545723478932277445296628684658831045668721
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72517863508546378037986835847163803922979010945310978933063021551647328425708
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37321507960574191463821484606429089071924140146207291159456579920511458395348
19857359070596142199415320243671339999379786502159576429167050778999850377372
17430038014325420655982788442390025163361003630125096695617319527983641501000
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97720893779694409008513485174792238681549493721905184119152068800385977707648
11654124570226643149438672777560976315460029526522227543273105677859036625567
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80333845328902689565979289066601789447755540016354127395103372643900132001630
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22868771239790327258115820525700306520801504602699718128230740213798176451623
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06041805906932793084892590880479213992858142578348156914103846101686771556088
97323903368726147845130022019573079364039513428786840754670553147660543544653
31492515634230596882192571220150357568620524049165708830104251833156693723642
35716159057124607255135672468657421157883801873562821756421821918147011583881
83758215647874478361346906736143470543403216693064026744491033069624005631000
55532461002156724154584664048053108207035047437804943564209482815492860783839
28627519203826947985905766509152756130661727289121399280207054974418108449943
90334436971429019536429813889894021700457087521572254378039371844320028813912
60082172784773738003588611520783053487764835555502533891652922516522011642219
30618470995094281212862801161435680719829105840478066391594453959494663629723
05764738570878837817341999144572624054052103979526915967861063507494118123083
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95347920597759474681905188339591339378201615082732079465134194186409827292427
35425666840873001188371760071948336532654757716096500625654477430676536914501
30354744089350676823728822794921768219692623910642685079816241637668383650805
99669160852595664720399406175116217711962710941315595616044451026595534340186
81041354663381952614563089084889234095193968662717118693279711358464919718882
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76212929705260616141472120305847552401983389590389314082114526577437000229416
94712286907025732828375081818468950122957597866484248365772567467169972576935
70927487681952882275029251083029726726852180626766538111289243621533056582657
25547177614055902219584094785214988781505088337257634920635362489979681584387
38932912323646807855228390505676831454576274354699273308660128568986008196832
66231194501080400462232955696145411127311954858278375762265340460852921456227
71477101948922930689919813688266588638251677298581401398394582716474854882636
40380828942703824737142205136944916615749488535096060173706149598542977566654
31994228440128874702953676303844744807656285831212984360940109051686265474987
86028591676495057944226320102114714256705877060558883127494842820187065077216
78332009781720649615433356208013886494947484990883131731493373260782697544005
85008742065093772217950386248701627128187737252584447050153677911791374061868
38999270660507990684216945707850002712159832205098105216679507189826185444478
85575951463619720926097358675086304302950713329793846219501530743707024059178
06452917124291184704293715612975429878700283114360378911152497575451374195073
42937124605548511058901401923472996435046697440475199665786644455962377624075
96543412794915218354390930683133468746288618050545813014681423560622989350409
73877941970079765248675693184563907233721930546544435780121391211507031601324
41333244710952855918489229283602220184033030300243697418918801752874208730003
46929873192869054502725378655908732398499773165181719523233495110938008695233
94338014556009913763537673753281956883664325045987054449239184336702048692820
02652687716829338746670971107634456331013950249009862946198731855543944052270
46060026336192154911469261083730629673759490986657831272704608083962361164417
02899512086846411923412208447340672713452206818245168896031096307620250340268
28407036053840225937209285485788708800932116965970491010037087592947667273329
4217057199083822060121