Robert P. P. McKone: As for the inclusion of this sequence, I do find this interesting because it shows the growth of the simplest recursive factorial.

Alois P. Heinz: citing Andrew: "This sequence is really not suitable for inclusion in oeis."

Robert P. P. McKone: Using A046969 to make it more accurate as the terms, I believe that the a-file value for a(4) while approximate, is probably the correct value, which can be checked with this altered Mathematica code:
approxA046969[n_,acc_]:=1+Floor[(1/Log[10])*(Log[Sqrt[2 Pi n]]+n*Log[n]-n+Log[1+Total[Table[1/(Sign[Numerator[BernoulliB[j]/((j-1)*j)]]*Denominator[BernoulliB[j]/((j-1)*j)]*n^(j-1)),{j,2,acc*2,2}]]])];
approxA046969[720!,100]

Robert P. P. McKone: I am always surprised by how accurate these approximations really are.

Robert P. P. McKone: I have moved this back to edit until I get back in front of a computer again to make an a-file for a(4).

Andrew Howroyd: b-file terms should not exceed 10^1000. See https://oeis.org/SubmitB.html ("Remember that the numbers cannot have more than about 1000 digits, or some of the programs will fail.")
This sequence is really not suitable for inclusion in oeis.

Andrew Howroyd: Having unique terms is not the criteria for whether a sequence is suitable for inclusion. This is not a database of every conceivable integer sequence. The definition itself is extremely marginal (not of general interest). In addition sequences that grow too rapidly like this have little to no search value.

Kevin Ryde: A b-file is strictly correct.  An a-file is anything you want (interesting/relevant/etc).

Kevin Ryde: If I have my factorials vs gamma the right way around, then I believe Ceiling[LogGamma[720! + 1]/Log[10]].  I thought mathematica automatically does what's needed to be certain of such value.

Robert P. P. McKone: I have two methods now to approximate this:
Ceiling[LogGamma[720!]/Log[10] + 1]
or
Floor[720!*Log10[720!] - 720!*Log10[E] + (Log10[2*Pi*720!])/2 + 1]
I do not have the mathematical knowhow to know which is better for larger numbers, the second one appears to be more accurate at smaller numbers, but I am not sure.  Also, I cannot find anything on what the OEIS says on what to do about adding approximations to a b-file?

Kevin Ryde: (Oh, I'm unclear there, I only mean lngamma(x) = log(gamma(x)) where log(y) is the natural logarithm of y.  I could have said the latter in the first place!  But the former was how I tried it. :)

Robert P. P. McKone: Firstly, I did make a mistake, a(4) is not too large to record, just too hard to calculate exactly.  As for the approximation, a(3) ~ Ceiling[LogGamma[720]/Log[10] + 1 // N] ~ 1745, compared to the known a(3) = 1747.  I can include an approximation of a(4) in an attached file?  But I have no idea what the error would be:
a(4) ~ 4541678546470819571014086338889057902415669057152237524509259157318352
1508848754381208076268912701252659700365051344686968203170966831918361
0818774548917227544170937273440112278637561075213115289182462181929849
5387314406979868225329388299905593419046942325721076076881401037736857
3757229564481516995810988152599989259185830267863548739483439218363522
8306922705638983169864165809862367506848732037491604837277664642661062
7414422899652228640611380333295977732537904423635687578617296579767260
9402726148770094132324366405621152401084496420186669013831338670541690
4141368572537381430038726562905439812451941585890503709699611267332679
2345857952799887773568923265957457054268883239161965867093317977041070
8917150466253682670326745320375551243573440345626806910596473652622437
1408416304884180100390648327553682846013103855492210459346312628288394
9560226538609600598143504165881213976891961817055029822494558310097940
0609234452192751805628230056162619939658586073969231394639033302975043
7462466329947995464471761176845193060644800714184778785769218923915818
7641428661713532019987947470878933290212604251376958660401990765491444
7057039774508057103769810384460004341320058909934996215776971610081108
8995901431791624100495414279124141824348455423686272292272854828861267
8531769195783198818168211428341123847359330616783052527607166807548113
2962387738570008812608216530013431836584106279235178808983305126595384
0616698198561499774618980155541484114089968122666350656177376678406473
1995356538362569270235296743593626774223344191622723624496341564657049
1841010118642650160299169499778356989118772668723891335822072973789735
7029675611643072246870360379220100891977116604860634227824346949014840
3930277094649365332044600566623227557639763838712800939884209621696512

Kevin Ryde: If I'm wrong, I'm missing where.  I'm looking at a(n) = ceil(lngamma(b(n-1)+1)/log(10)), yes?  No need to ever calculate b(n) to know its a(n) length, yes?

Kevin Ryde: (That's lngamma(x) in pari style, natural log log(gamma(x)).  As I mentioned, calculated without ever needing full value of gamma(x).)

Alois P. Heinz: yes ... correct ...