ConwayDigits[n_]:=
FindRoot[x^71-x^69-2*x^68-x^67+2*x^66+2*x^65+x^64-x^63-x^62-x^61-x^60-
x^59+2*x^58+5*x^57+3*x^56-2*x^55-10*x^54-3*x^53-2*x^52+6*x^51+6*x^50+
x^49+9*x^48-3*x^47-7*x^46-8*x^45-8*x^44+10*x^43+6*x^42+
8*x^41-5*x^40-12*x^39+7*x^38-7*x^37+7*x^36+
x^35-3*x^34+10*x^33+x^32-6*x^31-2*x^30-10*x^29-3*x^28+2*x^27+
9*x^26-3*x^25+14*x^24-8*x^23-7*x^21+9*x^20+3*x^19-4*x^18-10*x^17-7*x^16+
12*x^15+7*x^14+2*x^13-12*x^12-4* x^11-2*x^10+5*x^9+x^7-7*x^6+7*x^5-4*x^4+12*x^3
-6*x^2+3*x-6,{x, 2}, AccuracyGoal ->n, WorkingPrecision->n+50,MaxIterations->50]
Here is a value to 1000 digits.
{x -> 1.3035772690342963912570991121525518907307025046594048757548613906285\
50887852461557126815766864425225553471393047094902683962849893551554347375824\
85669108897777021657600666661361819575814997141621174776792313242993257201430\
46277132953880169252753163112124160278974455631122119289176594544284422105034\
83984430484350365978861636568696122933005217967812127420727428338418585389780\
46297536211133257131559611428416463649306506851827429249157508076047165548732\
89903140679626277521098230185009804867680417112703692827704555699254104725325\
52960023967416151761300757699523941470920962729779652528381289529239984214664\
69700636280871859847376432425963753778614871491804177943882564549629612672676\
31703413998923868380017499339163832550419738743376404644230963416377342999964\
02795347781231503693811136196919310110551222432806533487294985163535018000861\
89800442695696203908949639604105938945262628974848057200498952058940831545115\
27464184151251802473020019379677783662140660911047529322875883638588769569619\
9420516499767643762266466055465244027494584094408262251386608655407221`1050}
To see more details see the excellent account that
Steve Finch has done about the Conway constant.