Hi everyone,
here is another update. My computer has been calculating for a couple of days now and here is what I've got:
bin = Databin["zR0oyN8c"]
results = SortBy[DeleteDuplicates[Flatten["Data" /. Normal[Get@bin], 1]], First]
Grid[Join[{{"base", "# sequences", "max length", "string"}}, results],Frame -> All]

As you can see only the bases 24 and 30 are missing. Here is a graph of the behaviour - with these two bases missing:
ListLinePlot[results[[All, {1, 3}]], PlotTheme -> "Marketing", FrameLabel -> {"base", "max length"},
LabelStyle -> Directive[Bold, 16], ImageSize -> Large]

and
ListLogPlot[results[[All, {1, 2}]], PlotTheme -> "Marketing", FrameLabel -> {"base", "number strings"},
LabelStyle -> Directive[Bold, 16], ImageSize -> Large, PlotRange -> All, Joined -> True]

There are several observations here. Here is a little diagram with highlights:
ListLinePlot[results[[All, {1, 3}]], PlotTheme -> "Marketing",
FrameLabel -> {"base", "max length"},
LabelStyle -> Directive[Bold, 16], ImageSize -> Large,
PlotRange -> {All, {0, 75}},
Epilog -> {Red, Thickness[0.01], Line[{{0, 6}, {30, 73}}], Green,
Line[{{0, 0}, {35, 21}}], White, Opacity[0.4],
Rectangle[{12.5, 8}, {17.5, 28}],
Rectangle[{18.5, 10}, {23.5, 40}],
Rectangle[{24.5, 18}, {29.5, 48}],
Rectangle[{30.5, 19}, {35.5, 53}]}]

The green and red lines are to guide the eye and show estimates for lowest and highest values expected. The grey boxes indicate an M like structure. Left and right of which we expect particularly long sequences. The two values that are missing at 24 and 30 are exactly between two if these M like structures. So particularly high values occur at bases that are multiples of 6.
If this type of linear interpolation was correct we might expect string lengths of 58-60 for base 24 and something close to length 70 for base 30. It appears that the CPU time needed increases roughly exponentially with the length of the sequence, so we might have to wait for quite a bit for results on these bases.
Members of this Community have contacted me and have started to use more sophisticated methods to tackle these bases. I am convinced that they will beat me to it. I would love to see their results.
Also, I think that we all use PrimeQ. This uses heuristics for large prime numbers. It might be necessary to check whether these results are actually correct.
I have not found a list of left truncatable primes that has bases as large as the ones discussed here, so it is difficult to compare.
Also, we might want to start implementing something to be able to compute this for bases larger than 35 and perhaps think about mathematical arguments for the multiples of 6 observation. One thought is that this behaves a bit like the number of divisors of the respective base.
Cheers,
Marco