There are short runs of repeating digits in many, if not all, other bases like shown if the following examples.

Table[x = 11^n;
 l = x + Rationalize[RandomReal[WorkingPrecision -> 200]];
 For[a = 1, a < 80, c = Convergents[l, 110];
  l = FromContinuedFraction[c], a++];
 Print[BaseForm[N[l, 100], 11]], {n, 0, 10}]

gives

Subscript[1.555555555555555555555555555555555533700a2550688908575490a69364603964429224586627a93a3402a20a768, 11]

Subscript[10.0a9058978520259a142827a922a92569656863a36698a4338868409471057a972647425325419330582897044190a86, 11]

Subscript[100.00aaa90005aa88a9a0146210501696393768358512440139357a352a10a544892343a8259a40799127313463772096, 11]

Subscript[1000.000aaaaa9000005aaaa88aa90a00186629a204660340816aa4034a8a64332475161211097a661662a93a4a2685a02, 11]

Subscript[10000.0000aaaaaaa900000005aaaaaa88aaa900a00018a662a9a020460612040a81231a293588695452177a88a9172284, 11]

Subscript[100000.00000aaaaaaaaa90000000005aaaaaaaa88aaaa9000a000018aa662aa9a00204600612a040aa811951a28136015, 11]

Subscript[1.000000000000aaaaaaaaaaa9000000000005aaaaaaaaaa88aaaaa90000a0000018aaa662aaa9a0002046000612aa0410, 11]*11^(6)

Subscript[1.00000000000000aaaaaaaaaaaaa900000000000005aaaaaaaaaaaa88aaaaaa900000a00000018aaaa662aaaa9a000020, 11]*11^(7)

Subscript[1.0000000000000000aaaaaaaaaaaaaaa90000000000000005aaaaaaaaaaaaaa88aaaaaaa9000000a000000018aaaaa663, 11]*11^(8)

Subscript[1.000000000000000000aaaaaaaaaaaaaaaaa9000000000000000005aaaaaaaaaaaaaaaa88aaaaaaaa90000000a0000000, 11]*11^(9)

Subscript[1.00000000000000000000aaaaaaaaaaaaaaaaaaa900000000000000000005aaaaaaaaaaaaaaaaaa88aaaaaaaaa9000000, 11]*11^(10)

And

Table[x = 10^n;
 l = x + Rationalize[RandomReal[WorkingPrecision -> 200]];
 For[a = 1, a < 80, c = Convergents[l, 110];
  l = FromContinuedFraction[c], a++];
 Print[N[l, 100]], {n, 0, 10}]

gives

1.499999999999999999999999999999999999228053425371912510342130313447820081770198211507352907346219401

10.09805770662442796602740266883718035704414624140262096634681938285906167992862504291388332912688194

100.0099980005997601119434299835649973755091808365393805976599772401114132776432990139760009925703184

1000.000999998000005999976000111999434002999983556068423745679297941276947422877152236031026016426352

10000.00009999999800000005999999760000011199999434000029999998355599852424022296796516993457245096530

100000.0000099999999980000000005999999997600000001119999999434000000299999999835559997692424002706968

1.000000000000999999999998000000000005999999999976000000000111999999999434000000002999999999983556000*10^6

1.0000000000000099999999999998000000000000059999999999997600000000000111999999999994340000000000300000*10^7

1.0000000000000000999999999999999800000000000000059999999999999976000000000000011199999999999994340000*10^8

1.0000000000000000009999999999999999980000000000000000059999999999999999760000000000000001120000000000*10^9

1.0000000000000000000099999999999999999998000000000000000000059999999999999999997600000000000000000112*10^10

And

Table[x = 9^n;
 l = x + Rationalize[RandomReal[WorkingPrecision -> 200]];
 For[a = 1, a < 80, c = Convergents[l, 110];
  l = FromContinuedFraction[c], a++];
 Print[BaseForm[N[l, 100], 9]], {n, 0, 10}]

gives

Subscript[1.44444444444444444444444444444444444443073531882470086460233170312543468268784541455068201865618482513564, 9]

Subscript[10.0870564067366372663207648504632136088141106250228645515456237158105552460770327173468732711403618222108, 9]

Subscript[100.008887000588628832143270508682016054717823841110832851328460854107347124788711825234207688430331030438, 9]

Subscript[1000.00088888700000588886288713202123373664388052576487370184004252631020602411405156567187226134277084477, 9]

Subscript[10000.0000888888870000000588888862888701320021823387333643100875313652484784246074414317476421476832164160, 9]

Subscript[100000.000008888888887000000000588888888628888700132000218823388733036430101167531836412484528486860883365, 9]

Subscript[1.00000000000088888888888700000000000588888888886288888700013200002188823388873300364300101186753188364013, 9]*9^(6)

Subscript[1.00000000000000888888888888870000000000000588888888888862888888700001320000021888823388887330003643000101, 9]*9^(7)

Subscript[1.00000000000000008888888888888887000000000000000588888888888888628888888700000132000000218888823388888733, 9]*9^(8)

Subscript[1.00000000000000000088888888888888888700000000000000000588888888888888886288888888700000013200000002200000, 9]*9^(9)

Subscript[1.00000000000000000000888888888888888888870000000000000000000588888888888888888862888888888700000001320000, 9]*9^(10)