From a quick play around on Google Maps, these are the top results (that I could find):
ANE:
X18 Berwick to Newcastle - 151
22 Durham to Sunderland - 99
1 Darlington to Tow Law - 97
24 Hartlepool to Durham - 96
23 Hartlepool to Sunderland - 96
6 Barnard Castle to Framwellgate Moor - 91
GNE:
35A Boldon to Low Moorsley - 114
35 Low Moorsley to Boldon - 110
50 Marsden to Durham - 103
20 Langley Park to Sunderland, Winter Gardens - 99
10 Newcastle to Hexham - 83
21 Brandon to Newcastle - 78
SNE:
685 Carlisle to Newcastle - 109
59 Hartlepool to Durham - 81
6 Darlington to Stockton - 78 (stat does not include the Darlington to Hurworth portion)
36 Hartlepool to Middlesbrough - 68
1 High Tunstall to Middlesbrough - 68
Stats are to the best of my knowledge of routes when put into Google Maps where journey planner tells you to "ride" for however many stops. Number of stops shown is in the direction with the most stops.
Bus Route With The Most Stops