Brachylog, 7 6 5 bytes

ℕ₁≜¬∈

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-1 stealing Fatalize's solution to the dupe target--note that my accepted golf suggestion fails on the empty list. Whoops

Reversed I/O.

ℕ₁       The output is a positive integer.
  ≜      Try every value for it until one
   ¬∈    isn't in the input.

Brachylog, 7 6 5 bytes

ℕ₁≜¬∈

Try it online!

-1 stealing Fatalize's solution to the dupe target--note that my accepted reverted golf suggestion fails on the empty list. Whoops

Reversed I/O.

ℕ₁       The output is a positive integer.
  ≜      Try every value for it until one
   ¬∈    isn't in the input.

Brachylog, 7 6 bytes

,0≡ⁿℕ₁

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Seems not to need an explicit label. ~k≠tℕ₁ ties (see history for explanation), but only assuming the input contains no duplicates.

    ℕ₁    The output is a positive integer
  ≡ⁿ      which is not an element of the input
,0        (with 0 appended, because ⁿ doesn't like empty lists).

Brachylog, 7 6 5 bytes

ℕ₁≜¬∈

Try it online!

-1 stealing Fatalize's solution to the dupe target--note that my accepted golf suggestion fails on the empty list. Whoops

Reversed I/O.

ℕ₁       The output is a positive integer.
  ≜      Try every value for it until one
   ¬∈    isn't in the input.

Brachylog, 7 bytes

d~k≠tℕ₁

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Seems not to need an explicit label to function properly.

 ~k        Append something to
d          the deduplicated input
   ≠       such that all elements are distinct
    t      and the new last element
     ℕ₁    is a positive integer
           which is output.

Brachylog, 7 6 bytes

,0≡ⁿℕ₁

Try it online!

Seems not to need an explicit label. ~k≠tℕ₁ ties (see history for explanation), but only assuming the input contains no duplicates.

    ℕ₁    The output is a positive integer
  ≡ⁿ      which is not an element of the input
,0        (with 0 appended, because ⁿ doesn't like empty lists).