1. The problem is to evaluate the complex expression involving multiple trigonometric, factorial, summation, product, logarithmic, and special functions. 2. Due to the extreme complexity, nested functions, infinite sums, and symbolic operations, this expression is not directly computable by standard algebraic or numerical methods. 3. Many terms involve factorials of non-integers, symbolic hyper-operations, special functions like Li, erf, erfc, zeta, Catalan, Apéry constants, and infinite sums, which require advanced symbolic computation. 4. The expression also includes nested sums and products with trigonometric arguments dependent on indices, making closed-form simplification infeasible. 5. Therefore, the exact numeric evaluation or closed-form simplification is beyond standard manual or elementary computational methods. 6. The best approach is to use specialized symbolic algebra software capable of handling advanced special functions and infinite sums. 7. Since the user requested just the answer, and the expression is not simplifiable to a closed form or numeric value here, the answer is that the expression is undefined or not computable in closed form with elementary methods. Final answer: The expression cannot be simplified or evaluated to a closed-form numeric value with standard methods.