The answer is 1776.

Proof:

Define an operation a**b = ab+a+b

Clearly ** is commutative, because a**b = ab+a+b = ba+b+a = b**a

What's not so obvious is that ** is associative, at least until you realize that

a**b =
(a+1)(b+1)-1

Then (a**b)**c =
((a+1)(b+1))(c+1)-1 = (a+1)((b+1)(c+1))-1 = a**(b**c)

Since ** is associative and commutative, the 1775 operations can be performed in any order, using each of the 1775 operands, and the result will be the same. The number we always arrive at after performing 1775 operations is:

1**(¹/₂)**(¹/₃)** ... **(¹/₁₇₇₆), which is

(²/₁)(³/₂)(⁴/₃)...(¹⁷⁷⁷/₁₇₇₆)-1,
which is equal to 1777-1 = 1776 after everything else cancels.