Saturation of the core depends on the core material but usually for ferrite type materials it is round about 0.4 teslas (an example is 3C90 material from ferroxcube): -
Notice that despite an increasing H field (basically amps), the flux density is leveling out at about 400 mT. So what is the H field all about?
The H field is the thing that makes magnetism and for a regular ferrite (say a toroid), H field is amperes x turns / distance around toroid. By distance around toroid I means \$\ell_e\$ (effective length) shown below: -
So, a bigger toroid has a longer \$\ell_e\$ and therefore the H field for a given number of ampere-turns will be smaller. This means the toroid will saturate less for the same current.
What does a gap do? A gap reduces the effective permeability of the core material and, remembering that B = \$\mu H\$, a smaller permeability also means a smaller flux density for the same current.
In summary, it is amps x number of turns that is the driving force (magneto motive force) but when taking into account the effective length of the toroid (or E core etc..), MMF maps to magnetic field strength H and this, coupled with the effective permeability of the core dictates saturation point of flux density B.
So a bigger ferrite with a gap is a must when avoiding saturation.
It's also worthy of note that many designer's aim is to have a certain value of inductance and that introducing a gap may halve the permeability. This halves the inductance yet, increasing the turns by \$\sqrt2\$ restores the inductance to its previous value. Inductance is proportional to turns-squared.
This tells you that you can obtain the same inductance from a gapped version of a core AND lower the saturation point. However, copper losses may dictate that there is no net gain in "efficiency" without going to a bigger core.
Another method is to choose two inductors of twice the value and put them in parallel. If you look at a power inductor's spec you will see that the saturation current for the inductor of the higher value is more than half the saturation current of the inductor with half the inductance.
Reference BH curves for various materials: -
Magnetization curves of 9 ferromagnetic materials, showing saturation.
Cobalt and Nickel are both used in ferrites. Information taken from here