Typical microresonators exhibit a large frequency spacing between resonances and a limited tunability. This impedes their use in a large class of applications which require a resonance of the microresonator to coincide with a predetermined frequency. Here, we experimentally overcome this limitation with highly prolate-shaped whispering-gallery-mode "bottle microresonators" fabricated from standard optical glass fibers. Our resonators combine an ultrahigh quality factor of 3.6 x 10(8), a small mode volume, and near-lossless fiber coupling, characteristic of whispering-gallery-mode resonators, with a simple and customizable mode structure enabling full tunability.