Aims: Planets in the mass range from 2 to 15 M⊕ are very diverse. Some of them have low densities, while others are very dense. By measuring the masses and radii, the mean densities, structure, and composition of the planets are constrained. These parameters also give us important information about their formation and evolution, and about possible processes for atmospheric loss. Methods: We determined the masses, radii, and mean densities for the two transiting planets orbiting K2-106. The inner planet has an ultra-short period of 0.57 days. The period of the outer planet is 13.3 days. Results: Although the two planets have similar masses, their densities are very different. For K2-106b we derive Mb=8.36-0.94+0.96 M⊕, Rb = 1.52 ± 0.16 R⊕, and a high density of 13.1-3.6+5.4 g cm-3. For K2-106c, we find Mc=5.8-3.0+3.3 M⊕, Rc=2.50-0.26+0.27 R⊕ and a relatively low density of 2.0-1.1+1.6 g cm-3. Conclusions: Since the system contains two planets of almost the same mass, but different distances from the host star, it is an excellent laboratory to study atmospheric escape. In agreement with the theory of atmospheric-loss processes, it is likely that the outer planet has a hydrogen-dominated atmosphere. The mass and radius of the inner planet is in agreement with theoretical models predicting an iron core containing 80-30+20% of its mass. Such a high metal content is surprising, particularly given that the star has an ordinary (solar) metal abundance. We discuss various possible formation scenarios for this unusual planet. The results are partly based on observations obtained at the European Southern Observatory at Paranal, Chile in program 098.C-0860(A). This paper includes data gathered with the 6.5 m Magellan Telescopes located at Las Campanas Observatory, Chile. The article is also partly based on observations with the TNG, NOT. This work has also made use of data from the European Space Agency (ESA) mission Gaia (http://https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, http://https://www.cosmos.esa.int/web/gaia/dpac/consortium).The RV measurements are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (http://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/608/A93