APL (Dyalog Extended), 15 bytes

-4 thanks to dzaima and ngn

{{⍎4⍴⍕⍵*2}⍣⍵⊢1}

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Explanation:

{{⍎4⍴⍕⍵*2}⍣⍵⊢1}
 {⍎4⍴⍕⍵*2}⍣⍵       ⍝ apply the dfn ⍵ times
             1     ⍝ .. with initial argument 1
      ⍵*2          ⍝ current value squared
  ⍎4⍴⍕             ⍝ take first 4 digits

APL (Dyalog Extended), 13 bytes

-2 (changing the input method) thanks to @Razetime

{⍎4⍴⍕⍵*2}⍣⎕⊢1

Explanation:

{⍎4⍴⍕⍵*2}⍣⎕⊢1     ⍝ apply the dfn input times
            1     ⍝ .. with initial argument 1
     ⍵*2          ⍝ current value squared
 ⍎4⍴⍕             ⍝ take first 4 digits

Try it online!

APL (Dyalog Extended), 15 bytes

-4 thanks to dzaima and ngn

{{⍎4⍴⍕⍵*2}⍣⍵⊢1}

Try it online!

APL (Dyalog Extended), 15 bytes

-4 thanks to dzaima and ngn

{{⍎4⍴⍕⍵*2}⍣⍵⊢1}

Try it online!

Explanation:

{{⍎4⍴⍕⍵*2}⍣⍵⊢1}
 {⍎4⍴⍕⍵*2}⍣⍵       ⍝ apply the dfn ⍵ times
             1     ⍝ .. with initial argument 1
      ⍵*2          ⍝ current value squared
  ⍎4⍴⍕             ⍝ take first 4 digits

APL (Dyalog Extended), 19 bytes

{{⍎4↑∊⍕¨4⍴⍵*2}⍣⍵⊢1}

Try it online!

APL (Dyalog Extended), 15 bytes

-4 thanks to dzaima and ngn

{{⍎4⍴⍕⍵*2}⍣⍵⊢1}

Try it online!