Here are all the numbers which have not been obtained already by trolley813, Omega Krypton and ppgdev
$28 = 1 \times ((\sqrt{9})! \times 6) - 8$
$30 =-(1\times(\sqrt{9})!) + \sqrt{\sqrt{6^8}} $
$33 = 19 + 6 + 8$
$34 = ((1+(\sqrt{9})!) \times 6) - 8 $
$35 = (1 \times \sqrt{(\sqrt{9})^6}) + 8$
$36 = (1 + \sqrt{(\sqrt{9})^6}) + 8$
$37 = 1^9 + \sqrt{\sqrt{6^8}}$
$38 = (1+\sqrt{9})! + 6 + 8$
$41 = -(1 + (\sqrt{9})!) + (6 \times 8)$
$42 = -(1 \times (\sqrt{9})!) + (6 \times 8)$
$43 = -1 + ((\sqrt{9})! \times 6) + 8$
$44 = 1 \times ((\sqrt{9})! \times 6) + 8$
$45 = 1 + ((\sqrt{9})! \times 6) + 8$
$ 55 = 1 + (\sqrt{9})! + (6 \times 8)$
$56 = (1^9 + 6) \times 8 $
$ 64 = ((-1 + \sqrt{9}) + 6) \times 8$
$65 = -(1 \times \sqrt{9}) + 68 $
$66 = 1 - \sqrt{9} + 68 $
$69 = 1^9 + 68$
$70 = -1 + \sqrt{9} + 68 $
$71 = -1 + ((\sqrt{9} + 6)\times 8)$
$72 = ((1 \times \sqrt{9}) + 6) \times 8$
$73 = 1 + ((\sqrt{9} + 6)\times 8) $
$74 = (1 \times (\sqrt{9})!) + 68$
$75 = 1 + (\sqrt{9})!) + 68$
$ 79 = -1 + ((9 + \sqrt{\sqrt{\ldots \sqrt{6}}}) \times 8)$
$ 86 = -1 - \sqrt{9} + (6!/8)$
$90 = (1^9 \times 6!)/8 $
$91 = 1^9 + (6!/8)$
Omega Krypton had some of the answers with $6$ and $8$ switched. (thanks to 3D1T0R for spotting this) Here are those fixed which are not covered already by trolley813
$46 = 1 + 9 + \sqrt{\sqrt{6^8}}$
$47 = -1^9 + (6\times 8) $
$48 = 1^9 \times 6 \times 8$
$49 = 1^9 + (6\times 8) $
$50 = ((1 + (\sqrt{9})!) \times 6) + 8 $
$51 = (1 \times \sqrt{9}) + (6 \times 8) $
$ 53 = -1 + (\sqrt{9})! + (6 \times 8) $
$54 = (1 \times (\sqrt{9})!) + (6 \times 8) $
$61 = -1 - (\sqrt{9})! + 68$