dc, 46 bytes
[[{}]]sx256?^dd3^8d^1-/8092541**r255/BF*+d0=xP
Try it online!
Input on stdin, output on stdout.
This works by computing a formula for the desired output as a base-256 number. The P command in dc is then used to print the base-256 number as a string.
Further explanation:
Let n be the input n. The dc program computes the sum of
A = floor(256^n / 255) * 125 (BF is interpreted by dc as 11*10+15 = 125)
and
B = floor((256^n)^3 / (8^8-1)) * 8092541 * (256^n).
For A:
Observe that 1 + 256 + 256^2 + ... + 256^(n-1) equals (256^n-1)/255, by the formula for a geometric progression, and this equals floor(256^n / 255). So this is the number consisting of n 1's in base 256.
When you multiply it by 125 to get A, the result is the number consisting of n 125's in base 256 (125 is a single digit in base 256, of course). It's probably better to write the digits in base 256 as hex numbers; 125 is hex 7D, so A is the base-256 number consisting of n 7D's in a row.
B is similar:
This time observe that 1 + 16777216 + 16777216^2 + ... + 16777216^(n-1) equals (16777216^n - 1)/16777215, and this equals floor(16777216^n/16777215).
Now, 256^3 = 16777216, and 8^8-1 = 16777215, so this is what we're computing as floor((256^n)^3 / (8^8-1)).
From the geometric series representation, this number in base 256 is 100100100...1001 with n of the digits being 1 and the rest of the digits being 0.
This is multiplied by 8092541, which is 7B7B7D in hexadecimal. In base 256, this is a three-digit number consisting of the digits 7B, 7B, and 7D (writing those digits in hex for convenience).
It follows that the product written in base 256 is a 3n-digit number consisting of the 3 digits 7B 7B 7D repeated n times.
This is multiplied by 256^n, resulting in a 4n-digit base-256 number, consisting of the 3 digits 7B 7B 7D repeated n times, followed by n 0's. That's B.
Adding A + B now yields the 4n-digit base-256 number consisting of the 3 digits 7B 7B 7D repeated n times, followed by n 7D's. Since 7B and 7D are the ASCII codes for { and }, respectively, this is the string consisting of n copies of {{} followed by n copies of }, which is exactly what we want for n > 0. The P command in dc prints a base-256 number as a string, just as we need.
Unfortunately, n=0 has to be treated as a special case. The computation above happens to yield a result of 0 for n=0; in that case, I've just hard-coded the printing of the string {}.