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The Hierarchy Calculator

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For inputs like ( f_\omega+1(4) ), the output is astronomically large (beyond power towers). A high-quality calculator does attempt to print 10^10^... digits. Instead, it outputs: fast growing hierarchy calculator high quality

• Use tail recursion where possible.
• Implement iteration explicitly.
• For large ( n ), just compute the growth rate rather than exact number (e.g., “( f_3(4) ) is about ( 2 \uparrow\uparrow 4 )”).
• For pure display, stop after a certain tetration level and use arrow notation.