How massive can a cold solid object be? I understand that a neutron star is dense enough that adding more matter will increase the amount of degenerate matter directly, and the limit to its size is about 1.4 solar masses.

But if an object were formed not by an explosion that crushes it down to neutron matter, is it necessarily so?

Given normal atoms of any desired species, and it is piled up carefully so it does not get hot. When it gets around one solar mass, will it necessarily collapse into a neutron star, or can it get more massive?  Will such an object eventually collapse catastrophically or can it have a degenerate matter core and a substantial thickness of less-compressed matter on top and normal matter on top of that?

What is the largest possible mass of a cold solid object?  Can it be several solar masses?

By cold I mean that it’s not a star, puffed up by ongoing consumption of energy. Whatever it's made out of is not being “used up”.
The limit to the neutron star mass is at least $2M_{\odot}$, since at least two have measured masses as large as this (e.g. Demorest et al. 2010).

The answer to your question is the same as the answer to what is the maximum mass of a neutron/quark star, since if you compress matter of any sort, this is ultimately what it will become.

The answer to this question is also unknown and depends on the uncertain equation of state (the relationship between pressure, density and composition) at ultra-high densities, but must be somewhere between the $2M_{\odot}$ I mentioned above and an upper limit of around $3.5M_{\odot}$, which is imposed by General Relativity and an equation of state where the speed of sound equals the speed of light (the hardest possible equation of state). 

Whether these highly compressed objects are "solid" is also a matter of debate and research. The conventional picture of an old-ish white dwarf is that it is indeed a crystalline solid. The bulk of the interior of neutron stars of moderate mass is almost certainly a fluid of neutrons along with some protons and electrons. The neutrons may solidify at very high densities and this is thought to be one of the "harder" options for the equation of state.

To answer a comment by @zephyr - cold in this context means that the Fermi energies of degenerate fermion species are very much higher than $kT$. White dwarfs and neutron stars (older than a few seconds) are "cold", despite having interior temperatures, in the case of the latter, of a billion degrees. Making them even colder does not change the size/density of the object but does enable the interior of a white dwarf to crystallise.