The smallest carbon nanotubes with the chiral index (3,3), (4,3), or (5,1) were unambiguously identified for the first time. They were grown inside single-wall carbon nanotubes with the diameter of 1.0-1.2 nm, and the chiral indices were experimentally assigned beyond a doubt by an aberration-corrected high-resolution transmission electron microscopy (HR-TEM). In contrast to a theoretical prediction, the (3,3) nanotube is rather unstable and extremely sensitive to the electron beam and, therefore, may not survive alone without the protection of outer nanotube. The cap structure of (3,3) nanotube is also well-explained by a half-dome of C20 fullerene, which consists of six pentagons only.