Approximate the Dottie number
MATL, 34 30 19 bytes
11 bytes off thanks to Sanchises!
48i:&quot;'cos('wh41hGY$

The last decimal figures in the output may be off. However, the number of correct figures starting from the left increases with the input, and the result converges to the actual constant.
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Explanation
For input n, and starting at x=1, this applies the function
              x ↦ cos(x)
with n-digit variable-precision arithmetic n times.
48         % Push 48, which is ASCII for '1': initial value for x as a string
i:&quot;        % Do n times, where n is the input
  'cos('   %   Push this string
  w        %   Swap. Moves current string x onto the top of the stack
  h        %   Concatenate
  41       %   Push 41, which is ASCII for ')'
  h        %   Concatenate. This gives the string 'cos(x)', where x is the
           %   current number
  GY$      %   Evaluate with variable-prevision arithmetic using n digits
           %   The result is a string, which represents the new x
           % End (implicit). Display (implicit). The stack contains the last x