I would like to tentatively claim that 265 is the answer, since I think I did a pretty good go at trying to solve it, and haven't managed it yet.
Continuing from Dante, Weather Vane, BlueHairedMeerkat, and Mike's efforts. As with their answers, please comment or edit in any gaps you solve.
Gaps at: 265-267, 275-279, 281, 283, 293, 295, 297, and 303 onward
Like BlueHairedMeerkat's answer:
Note that 'precisely 4 1s and 4 0s' is equivalent to 'up to 4 1s and 4 0s', since we can trivially add 0 or multiply by 1 use up additional digits. These trivial digits have been omitted for readability.
$263 = 10^{100!!}+11!+0!$
$264 = 10^{100!!}+(11+0!)!!$
$265 = 101!/.1+100!+0!$
$266 = (101!+0!)/.1+100!$
$267 = (11!)!*.011-0!-0!-0!$
$268 = (11!)!*.011-0!-0!$
$269 = 101! \times 10.01-0!$
$270 = 101! \times 10.01$
$271 = 101! \times 10.01+0!$
$272 = (101!!)^{0!+0!}+(11!)!!-0!$
$273 = (101!!)^{0!+0!}+(11!)!!$
$274 = (101!!)^{0!+0!}+(11!)!!+0!$
$275 = (100!-0!)*11!/.1-0!$
$276 = (100!-0!)*11!/.1$
$277 = (100!-0!)*11!/.1+0!$
$278 = (11!+0!)!!/.011-0!-0!?$
$279 = (11!+0!)!!/.011-0!$
$280 = (11!)!/(0!+0!+.1)-100!!$
$281 = (11!+0!)!!/.011+0!$
$282 = (11!)!/(0!+0!+.1)-(10+0!)!$
$283 = ?$
$284 = (11!)!/(0!+0!+.1)-100$
$285 = (11!)!/(10.1)-0!-0!-0!$
$286 = (11!)!/(10.1)-0!-0!$
$287 = (11!)!/(10.1)-0!$
$288 = (11!)!/(10.1)$
$289 = (11!)!/(10.1)+0!$
$290 = (11!)!/(10.1)+0!+0!$
$291 = (11!)!/(10.1)+0!+0!+0!$
$292 = (11!)!/(0!+0!+.1)+100$
$293 = ?$
$294 = (11!)!/(0!+0!+.1)+(10+0!)!$
$295 = ?$
$296 = (11!)!/(0!+0!+.1)+100!!$
$297 = ?$
$298 = 101! \times 10.1-0!-0!$
$299 = 101! \times 10.1-0!$
$300 = 101! \times 10.1$
$301 = 101! \times 10.1+0!$
$302 = 101! \times 10.1+0!+0!$
$303 = ?$
