clearvars

for n= sym(50): sym(3000000)

  c=[char(string(n)) ...

    char(string(n^2)) ...

    char(string(n^3)) ...

    char(string(n^4)) ...

    char(string(n^5)) ...

    char(string(n^6)) ...

    char(string(n^7)) ...

    char(string(n^8)) ...

  ];

  ct=length(find(c=='2'|c=='3'|c=='8'));

  if ct==96

    disp([n ct])

  end

end

finds first (and once found, I stopped the program):

[1186022, 96]

so the answer is 1,186,022.

The powers are

1186022, 

1406648184484, 

1668315693058082648, 

1978659114912133298346256, 

2346733240786318158771223233632, 

2783277251703870635302163721998691904, 

3301028052620328058622342821892332569365888, 

3915091893024866724743388278306388058584469217536

The concatenation is 224 digits long.

In the vicinity of 1186022, the 96 occurrences is a high outlier. It's the only one that's 96 between 1186000 and 1187000. The first 50 counts in this area are

[1186000, 28]

[1186001, 69]

[1186002, 63]

[1186003, 77]

[1186004, 70]

[1186005, 71]

[1186006, 68]

[1186007, 66]

[1186008, 71]

[1186009, 67]

[1186010, 52]

[1186011, 61]

[1186012, 80]

[1186013, 59]

[1186014, 63]

[1186015, 75]

[1186016, 69]

[1186017, 62]

[1186018, 77]

[1186019, 51]

[1186020, 67]

[1186021, 67]

[1186022, 96] <-- the only 96

[1186023, 77]

[1186024, 64]

[1186025, 73]

[1186026, 69]

[1186027, 71]

[1186028, 62]

[1186029, 67]

[1186030, 73]

[1186031, 72]

[1186032, 78]

[1186033, 80]

[1186034, 66]

[1186035, 67]

[1186036, 66]

[1186037, 80]

[1186038, 73]

[1186039, 74]

[1186040, 55]

[1186041, 74]

[1186042, 67]

[1186043, 59]

[1186044, 62]

[1186045, 67]

[1186046, 68]

[1186047, 65]

[1186048, 66]

[1186049, 58]

[1186050, 62]

in keeping with what we might expect as 224 * .3 = 67.2

In contrast a different range of one thousand, higer up by an order of magnitude:

[21002023, 96]

[21002152, 96]

[21002221, 96]

[21002467, 96]

[21002823, 96]

[21002935, 96]

where the lengths are of the order of 267, and 267 * .3 = 80.1.

or

[31002131, 96]

[31002206, 96]

[31002207, 96]

[31002233, 96]

[31002265, 96]

[31002428, 96]

[31002532, 96]

[31002536, 96]

[31002631, 96]

[31002688, 96]

[31002787, 96]

[31002808, 96]

[31002853, 96]

[31002911, 96]

[31002984, 96]

where the concatenations are closer to 272 long and 272 * .3 = 81.6.

or to this span of 1000:

[331002031, 96]

[331002033, 96]

[331002047, 96]

[331002049, 96]

[331002056, 96]

[331002061, 96]

[331002064, 96]

[331002081, 96]

[331002120, 96]

[331002121, 96]

[331002225, 96]

[331002262, 96]

[331002338, 96]

[331002363, 96]

[331002424, 96]

[331002468, 96]

[331002536, 96]

[331002537, 96]

[331002552, 96]

[331002625, 96]

[331002628, 96]

[331002669, 96]

[331002724, 96]

[331002756, 96]

[331002763, 96]

[331002782, 96]

[331002802, 96]

[331002807, 96]

[331002814, 96]

[331002839, 96]

[331002852, 96]

[331002867, 96]

[331002896, 96]

where the concatenated lengths are over 310.

I'm sure that as the numbers get higher, eventually 96 will become an outlier as lower than the average altogether.