We report on the discovery of one of the most extreme cases of high-frequency radio variability ever measured in active galactic nuclei (AGNs), observed on time-scales of days and exhibiting variability amplitudes of 3-4 orders of magnitude. These sources, all radio-weak narrow-line Seyfert 1 (NLS1) galaxies, were discovered some years ago at Aalto University Mets & auml;hovi Radio Observatory (MRO) based on recurring flaring at 37 GHz, strongly indicating the presence of relativistic jets. In subsequent observations with the Karl G. Jansky Very Large Array (JVLA) at 1.6, 5.2, and 9.0 GHz no signs of jets were seen. To determine the cause of their extraordinary behaviour, we observed them with the JVLA at 10, 15, 22, 33, and 45 GHz, and with the Very Long Baseline Array (VLBA) at 15 GHz. These observations were complemented with single-dish monitoring at 37 GHz at MRO, and at 15 GHz at Owens Valley Radio Observatory (OVRO). Intriguingly, all but one source either have a steep radio spectrum up to 45 GHz, or were not detected at all. Based on the 37 GHz data, the time-scales of the radio flares are a few days, and the derived variability brightness temperatures and variability Doppler factors are comparable to those seen in blazars. We discuss alternative explanations for their extreme behaviour, but so far no definite conclusions can be made. These sources exhibit radio variability at a level rarely, if ever, seen in AGN. They might represent a new type of jetted AGN, or a new variability phenomenon, and thus deserve our continued attention.