thecages

Math::BigFloat does it after 128 iterations!
It turns out that normal float is correct up until the 51st iteration but then loses it on iteration 52.

Using Math::BigFloat is horrendously slow, though.

Let's brute a^(a+1) = (a+1)^a
Using BigFloater

It's somewhere between 2 and 3...
1 a= 2.5
2 a= 2.25
3 a= 2.375
4 a= 2.3125
5 a= 2.28125
6 a= 2.296875
7 a= 2.2890625
8 a= 2.29296875
9 a= 2.294921875
10 a= 2.2939453125
11 a= 2.29345703125
12 a= 2.293212890625
13 a= 2.2930908203125
14 a= 2.29315185546875
15 a= 2.293182373046875
16 a= 2.2931671142578125
17 a= 2.29315948486328125
18 a= 2.293163299560546875
19 a= 2.2931652069091796875
20 a= 2.29316616058349609375
21 a= 2.293166637420654296875
22 a= 2.2931663990020751953125
23 a= 2.29316627979278564453125
24 a= 2.293166339397430419921875
25 a= 2.2931663095951080322265625
26 a= 2.29316629469394683837890625
27 a= 2.293166287243366241455078125
28 a= 2.2931662909686565399169921875
29 a= 2.29316628910601139068603515625
30 a= 2.293166288174688816070556640625
31 a= 2.2931662877090275287628173828125
32 a= 2.29316628747619688510894775390625
33 a= 2.293166287359781563282012939453125
34 a= 2.2931662874179892241954803466796875
35 a= 2.29316628738888539373874664306640625
36 a= 2.293166287403437308967113494873046875
37 a= 2.2931662874107132665812969207763671875
38 a= 2.29316628741435124538838863372802734375
39 a= 2.293166287412532255984842777252197265625
40 a= 2.2931662874116227612830698490142822265625
41 a= 2.29316628741207750863395631313323974609375
42 a= 2.293166287411850134958513081073760986328125
43 a= 2.2931662874119638217962346971035003662109375
44 a= 2.29316628741190697837737388908863067626953125
45 a= 2.293166287411878556667943485081195831298828125
46 a= 2.2931662874118643458132282830774784088134765625
47 a= 2.29316628741185724038587068207561969757080078125
48 a= 2.293166287411860793099549482576549053192138671875
49 a= 2.2931662874118625694563888828270137310028076171875
50 a= 2.29316628741186168127796918270178139209747314453125
51 a= 2.293166287411861237188759332639165222644805908203125
52 a= 2.2931662874118610151441544076078571379184722900390625
53 a= 2.29316628741186112616645687012351118028163909912109375
54 a= 2.293166287411861070655305638865684159100055694580078125
55 a= 2.2931662874118610428997300232367706485092639923095703125
56 a= 2.29316628741186102902194221542231389321386814117431640625
57 a= 2.293166287411861035960836119329542270861566066741943359375
58 a= 2.293166287411861032491389167375928082037717103958129882812
59 a= 2.293166287411861030756665691399120987625792622566223144531
60 a= 2.2931662874118610316240274293875245348317548632621765136715
61 a= 2.2931662874118610311903465603933227612287737429141998291012
62 a= 2.2931662874118610314071869948904236480302643030881881713864
63 a= 2.293166287411861031515607212138974091431009583175182342529
64 a= 2.2931662874118610314613971035146988697306369431316852569577
65 a= 2.29316628741186103148850215782683648058082326315343379974335
66 a= 2.29316628741186103150205468498290528600591642316430807113617
67 a= 2.293166287411861031508830948560939688718463003169745206832585
68 a= 2.293166287411861031505442816771922487362189713167026638984378
69 a= 2.293166287411861031507136882666431088040326358168385922908482
70 a= 2.2931662874118610315079839156136853883793946806690655648705335
71 a= 2.2931662874118610315084074320873125385489288419194053858515593
72 a= 2.2931662874118610315081956738504989634641617612942354753610464
73 a= 2.2931662874118610315080897947320921759217782209816505201157899
74 a= 2.2931662874118610315080368551728887821505864508253580424931617
75 a= 2.2931662874118610315080103853932870852649905657472118036818476
76 a= 2.29316628741186103150802362028308793370778850828628492308750465
77 a= 2.293166287411861031508030237727988357929187479555821482790333175
78 a= 2.293166287411861031508026929005538145818487993921053202938918912
79 a= 2.293166287411861031508028583366763251873837736738437342864626044
80 a= 2.293166287411861031508027756186150698846162865329745272901772478
81 a= 2.293166287411861031508028169776456975360000301034091307883199261
82 a= 2.2931662874118610315080283765716101136169190188862643253739126525
83 a= 2.2931662874118610315080282731740335444884596599601778166285559568
84 a= 2.29316628741186103150802832487282182905268933942322107100123430465
85 a= 2.29316628741186103150802829902342768677057449969169944381489513072
86 a= 2.29316628741186103150802828609873061562951707982593863022172554376
87 a= 2.29316628741186103150802829256107915120004578975881903701831033724
88 a= 2.2931662874118610315080282893299048834147814347923788336200179405
89 a= 2.29316628741186103150802829094549201730741361227559893531916413887
90 a= 2.293166287411861031508028291753285584253729701017208986168737238055
91 a= 2.2931662874118610315080282913493888007805716566464039607439506884625
92 a= 2.2931662874118610315080282911474404090439926344610014480315574136662
93 a= 2.2931662874118610315080282912484146049122821455537027043877540510644
94 a= 2.29316628741186103150802829129890170284642690110005333256585236976345
95 a= 2.29316628741186103150802829127365815387935452332687801847680321041392
96 a= 2.29316628741186103150802829126103637939581833444029036143227863073916
97 a= 2.29316628741186103150802829125472549215405023999699653291001634090178
98 a= 2.29316628741186103150802829125157004853316619277534961864888519598309
99 a= 2.293166287411861031508028291249992326722724169164526161518319623523745
100 a= 2.2931662874118610315080282912507811876279451809699378900836024097534175
101 a= 2.2931662874118610315080282912511756180805556868726437543662438028682537
102 a= 2.2931662874118610315080282912509784028542504339212908222249231063108356
103 a= 2.29316628741186103150802829125087979524109780744561435615426275803212655
104 a= 2.29316628741186103150802829125083049143452149420777612311893258389277202
105 a= 2.29316628741186103150802829125080583953123333758885700660126749682309476
106 a= 2.29316628741186103150802829125081816548287741589831656486010004035793339
107 a= 2.293166287411861031508028291250812002507055376743586785730683768590514075
108 a= 2.293166287411861031508028291250808921019144357166221896165975632706804418
109 a= 2.293166287411861031508028291250807380275188847377539451383621564764949589
110 a= 2.2931662874118610315080282912508066099032110924831982289924445307940221745
111 a= 2.2931662874118610315080282912508062247172222150360276177968560138085584672
112 a= 2.2931662874118610315080282912508060321242277763124423121990617553158266136
113 a= 2.2931662874118610315080282912508059358277305569506496594001646260694606868
114 a= 2.2931662874118610315080282912508058876794819472697533330007160614462777234
115 a= 2.2931662874118610315080282912508058636053576424293051698009917791346862417
116 a= 2.29316628741186103150802829125080587564241979484952925140085392029048198255
117 a= 2.293166287411861031508028291250805869623888718639417210600922849712584112125
118 a= 2.293166287411861031508028291250805866614623180534361190200957314423635176912
119 a= 2.293166287411861031508028291250805865109990411481833180000974546779160709306
120 a= 2.293166287411861031508028291250805864357674026955569174900983162956923475503
121 a= 2.2931662874118610315080282912508058647338322192187011774509788548680420924045
122 a= 2.2931662874118610315080282912508058645457531230871351761759810089124827839538
123 a= 2.2931662874118610315080282912508058644517135750213521755384820859347031297284
124 a= 2.2931662874118610315080282912508058644046938009884606752197326244458133026157
125 a= 2.29316628741186103150802829125080586438118391397201492506035789370136838905935
126 a= 2.29316628741186103150802829125080586436942897046379204998067052832914593228118
127 a= 2.293166287411861031508028291250805864375306442217903487520514211015257160670265
128 a= 2.293166287411861031508028291250805864372367706340847768750592369672201546475722
15.38063255430049686075251318401086734804=15.38063255430049686075251318401086734804!