05AB1E, 11 bytes
Δγε2ôSO}˜J
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Explanation:
Δ # Continue until the (implicit) input no longer changes:
γ # Split the integer in chunks of the same adjacent digits
# i.e. 199999889 → [1,99999,88,9]
ε } # Map each to:
2ô # Split it into parts of size 2
# i.e. 99999 → [99,99,9]
€S # Split each part into digits
# i.e. [99,99,9] → [[9,9],[9,9],[9]]
O # And take the sum of each part
# i.e. [[9,9],[9,9],[9]] → [18,18,9]
˜ # Flatten the list
# i.e. [[1],[18,18,9],[16],[9]] → [1,18,18,9,16,9]
J # Join everything together
# i.e. [1,18,18,9,16,9] → 118189169
# (And output the result implicitly at the end)
# i.e. output = 28189169