Would an extremely slowly forming star ignite? Nuclear fusion requires extremely high temperatures and pressures, both of which are crated by the collapse of protostars.
But, what if the accretion of matter happened slowly enough that the core never got very hot.
For our example, we can take Jupiter, then very slowly (e.g. at a rate of say one Earth mass every few trillin years) would we be able to get it to 100 Jupiter masses without igniting fusion?
 A: Yes you could. But the mass increase rate would have to be sufficiently slow to allow the planet to cool at each stage. I think this would indeed mean significant mass increases would have to take place over trillions of years. 100 Jupiter masses would be possible, but there is a limit and ignition of "cold" fusion via pycnonuclear reactions should start at about 0.8 solar masses.
Details and background:
It isn't accretion that makes the interior hot. A non-accreting ball of gas will gradually contract as radiative losses extract energy from the star.
As the star contracts, the viral theorem ensures that half the released gravitational potential is radiated away and half is used to increase the kinetic energy (i.e. temperature of the gas). It is easy to show that for gas governed by perfect ideal gas pressure, that the temperature goes up as $M/R$.
Nuclear fusion will begin once the core temperature exceeds some threshold. The only way this can then be avoided is if the electrons in the gas become degenerate before the gas temperature reaches this threshold. The electron-degenerate gas pressure would become independent of temperature and the ball of gas could cool without significantly contracting any further.
Jupiter is almost in an electron-degenerate configuration. It will not now get much smaller or hotter in its interior as it cools and thus it won't commence fusion. If you added mass to Jupiter, it would stay at roughly the same size, but its interior temperature would grow and the luminosity (fuelled by the small amount of gravitational contraction) would also grow. At 13 times it's mass, its core would be hot enough to ignite the small fraction of deuterium in its interior. This would burn for about 100 million years at constant luminosity, and at a rate sufficient to stop it contracting. After that, if you kept adding mass then hydrogen fusion would start at about 75 Jupiter masses.
However, if you added the mass really slowly then you could try to create a rather peculiar kind of white dwarf. You would have to add the gas more slowly than the cooling timescale (which is indeed trillions of years). You could then build up the object through a series of "zero temperature" degenerate configurations of increasing mass and make what amount to a "hydrogen white dwarf" (assuming that what you added was mostly hydrogen). A hydrogen white dwarf is about 3 times the radius of a more typical white dwarf at the same mass, and therefore about thirtly times less dense (important later).
However, you still couldn't avoid nuclear fusion at some point. The increasing density (the density would increase as $M^2$ ) would reach $\sim 10^9$ kg/m$^3$ atabout 0.8 solar masses and this would be sufficient to start pycnonuclear reactions in hydrogen (Shapiro & Teukolsky 1983). It is possible that igniting the hydrogen in degenerate conditions would result in some type of supernova.
