Thwart Lepton compression
Python 3 + mozjpeg + /dev/urandom, 720×720: avg. score 102%
Depends on the mozjpeg package, the code assumes it's installed into /usr/local/opt/mozjpeg. (on OS X it's trivial to install, just run brew install mozjpeg)
Also it depends on /dev/urandom special file, it's used for generating random data.
The code simply feeds random data to mozjpeg compressor (in TGA format, because cjpeg understands it and it has very simple header), and lets it create an optimized jpeg file. Quality is set to the maximum because it makes DCT coefficients the least compressable, and it doesn't matter much what algorithm is used to compress uncompressable data.
I checked that the jpeg-&gt;lepton-&gt;jpeg cycle is lossless — it is true.
import subprocess
from subprocess import PIPE

c_mozjpeg_path = '/usr/local/opt/mozjpeg/bin/cjpeg'
cjpeg_params = '-quality 100 -dc-scan-opt 2 -dct float -targa'
image_size = 720


def write_random_tga_image(w, h, of, rf):
    def wb(value, size):
        of.write(int.to_bytes(value, size, 'little'))

    wb(0, 2)
    wb(3, 1)
    wb(0, 9)
    wb(w, 2)
    wb(h, 2)
    wb(8, 1)
    wb(0, 1)

    data_size = w * h
    while data_size &gt; 0:
        data_size -= of.write(rf.read(data_size))


def main():
    with open('/dev/urandom', 'rb') as rf:
        with open('oops.jpg', 'wb') as f:
            p = subprocess.Popen((c_mozjpeg_path,) + tuple(cjpeg_params.split(' ')), stdin=PIPE, stdout=f)
            write_random_tga_image(image_size, image_size, p.stdin, rf)
            p.communicate()


if __name__ == '__main__':
    main()

Code is not golfed, obviously.
Example image: 
Fun fact: generated JPEG file is larger than the source uncompressed TGA image, even though JPEG uses lossy compression.
Fun fact 2: Imgur (the default image hosting for SO) does very bad job at this file — for some reason it recompresses it to lower quality, even though it's less than 1MB. So I used Github to upload the example image.
Fun fact 3: In general, mozjpeg indeed does better JPEG compression while staying compatible with existing JPEG decoders. And it also has tool to losslessly optimize JPEG files, too — jpegtran.