The published honing rates
Tier 3 1340 gear hones at 100% through +6, then the published base rates fall: +7 at 60%, +8 at 45%, +9 through +11 at 30%, +12 through +14 at 15%, +15 through +17 at 10%, +18 and +19 at 5%, and +20 at 3%. Expected taps per level = 1 / rate, so a 10% level takes 10 taps on average and a 3% level takes 33.3.
How the totals are built
Each tap has a published honor shard and silver cost that rises with level. Expected cost = sum over every remaining level of (cost / rate). Averages are honest here because every tap is independent: no streak changes the next roll, except the artisan's energy pity meter, which this calculator does not model.
Worked example
+15 to +20 on 1340 gear. Rates: 10%, 10%, 5%, 5%, 3%. Expected taps = 10 + 10 + 20 + 20 + 33.3 = 93.3 taps. Expected honor shards = 47,987. Expected silver = 2,747,067. The last level alone averages 33 taps, which is why +19 to +20 feels endless.
What is not included
These totals cover honor shards and silver only. Each tap also consumes destruction or guardian stones, leapstones, and gold, whose amounts vary by piece and patch. Add those separately from your server's prices. Stronghold honing buffs and rate-up materials are also excluded, so buffed honing will beat these numbers.