How can I help SumConvergence give the right result?

The culprit is Sum`SumConvergenceDump`UnivariateLogarithm[], which mistakenly decides the sum is not convergent. It should be reported as a bug. (It would be acceptable if it couldn't decide, but to reach the wrong conclusion is wrong.)

Here's a modest implementation of the limit comparison test within the log-testing code. It uses the Villegas-Gayley trick to insert the code ahead of the built-in UnivariateLogarithm[] codes. We need to manually insert it as the first code in the DownValues, so that it is called before other definitions of UnivariateLogarithm[]. Since UnivariateLogarithm[] is buggy, it's a question whether I should call it (or SumConvergence[], which in turn would call it) after the comparison test to check convergence of the transformed series. I probably shouldn't unless I can prove I've avoided the bug, but just how much work should I do rooting around undocumented functions for free? Better to let WRI decide how to fix their software.

Internal`InheritedBlock[{Sum`SumConvergenceDump`UnivariateLogarithm},

 DownValues[Sum`SumConvergenceDump`UnivariateLogarithm] = Prepend[
   DownValues[Sum`SumConvergenceDump`UnivariateLogarithm],
   (* new def. for UnivariateLogarithm[] *)
   HoldPattern[
     Sum`SumConvergenceDump`UnivariateLogarithm[expr_, k_] /;
       ! TrueQ[$inLimitComparisionTestQ] && ! FreeQ[expr, _Log]
     ] :> Block[{$inLimitComparisionTestQ = True},
     Module[{factors, comparisons, log, nlogs, res},
      factors = Rest@FactorList[expr];
      nlogs = Max[Count[#, _Log, Infinity, Heads -> True] & /@ 
         factors[[All, 1]]];
      factors = Power @@@ factors;
      log = k; (* log is the iterated composition of Log[] 
                  with k up to nlogs number of times *)
      While[Depth[log] <= nlogs + 1 && ! TrueQ@res,
       comparisons = Abs@Limit[log*factors, n -> Infinity];
       res = Sum`SumConvergenceDump`UnivariateLogarithm[
         Times @@ ReplacePart[
           factors,
           Position[comparisons, L_ /; 0 < L < Infinity] -> 1/log],
         k];
       log = Log@log
       ];
      res /; TrueQ@res
      ]
     ]
   ];

 SumConvergence[1/(n Log[n] Log[n Log[n]]), n]
 ]

(*  True  *)