Density and radius are reliant on what a planet is made out of
AlexP's idea of planet made out of tungsten from the comments may be less "sexy", than gold but it is more along the right idea. There is no circumstance where only gold would accrete into a planet, but a planet that has spent a long time really close to a star can evaporate away all the lighter elements. The highest density you can get with a planet would be from one that averages about 4600-5000°C. This will boil away everything else leaving just a molten mass of Tungsten, Osmium, Rhenium, and Tantalum. If something were to then happen that pulls or pushes the planet farther away from the star, you would be left with a solid heavy metal world with a density of somewhere between 16.65-22.59 g/cm³ depending on the ratios of these 4 remaining elements. Since you won't get a purely Osmium world this way, your actual density cap is probably going to be somewhere around 20 g/cm³. (Technically a purely Rhenium planet could be 21 g/cm³ but its boiling point is so close to the less dense Tungsten that boiling off Tungsten without also losing your Rhenium is unfeasible).
Since Earth has a density of 5.51 g/cm³, this means your max density will be about 3.63 times that of Earth.
All together this means that planet at 0.3 earth radius would have 0.027 Earth volumes and a maximum mass of about 0.1 Earths and 1.11G.
If your goal is instead 4.44G you need a radius just a bit over 1.2 Earths.
A third (possible?) solution would be if this planet contained several times as many neutrons as normal matter. Since neutrons contribute mass, but no charge, you can just bind them to normal matter to increase its mass and density. Doing so would give you the dimensions you are looking for (with L. Dutch's corrections). That said, such a planet would become extremely radioactive. I'm not sure how to calculate at what point such a planet would simply become a giant nuclear bomb; so, I'm not 100% sure that this is actually viable, but it would probably be the most believable explanation of such proportions. Perhaps this scenario could be explained by the planet forming in the debris of an exploded neutron star.