Extreme currents aren't necessary; the core material is relevant, and can be switched at much lower currents depending on type.
The easiest way to experiment with core today is probably to find a magamp core in an oldish ATX power supply (they were used to separately regulate the 3.3V output from the 5V (AC) supply, and look like any old toroid choke, except for details about how they're wired). These are either ferrite, or amorphous/nanocrystalline strip, with a "square" B-H curve. Flux (volt-seconds) is only delivered in response to sufficient current flow to push the operating point around the hysteresis loop.
Magnetic core takes advantage of hysteresis, by using the core itself as a SUM gate, i.e. it only flips when N of the M inputs (read: wires through the center) are active. Which, when N = M, this is simply an AND function. So there's a row, column, and sometimes plane or bit line as well, which, only when all activated, flux is generated and thus a pulse detected (or not), successfully reading the core.
The core then needs to be remagnetized, of course (by the same process), to furnish a lossless read operation, that is (or perform a write while discarding the previous value).
Magamp cores are a poor choice, because they're optimized for minimum losses -- that is, minimum hysteresis loop area. The height of the loop is still about the saturation point (0.25-0.4T for ferrite, 0.8-1.5T for iron), which means the width is made low (i.e., few amp-turns required to switch it). Which is good for the application, that's less bias current required to drive it, less power dissipated. But you can still demonstrate core memory this way, scaling drive current appropriately; it's just more easily corrupted.
Conversely, hard magnets are made with maximal width (lots of amp-turns required to magnetize), which is good for them so they can magnetize things in turn (i.e., be a permanent magnet), and not good here because you need massive currents to flip even small cores.
If you must -- you can of course find some either vintage, or modern equivalents or substitutes (including perhaps magamp cores like this), and make a more authentic test circuit.
Oh, and to be clear -- "square" materials are in contrast to "soft" materials used for inductors and transformers, which have minimal hysteresis loss by reducing loop area along either axis -- this can sometimes lead to the same thing (some high-μ nanocrystalline materials for transformer service are nearly square*) but mostly leads to lower remenance. Since the zero-bias ("initial") slope (μ = dB/dH) is significant in these types, you don't really observe "switching" behavior.
*Which are likewise suitable for mag-amp use, but due to how narrow the loop is (coercion as low as a few mAt?), it may be quite difficult to actually demonstrate. Like I said, they may be a poor choice.
Ah, I should also mention the correct terms. Remenance is the flux density (B) at zero magnetization (H = 0). Coercion is the H required to force B to zero. On the B-H curve, coercion is the width (or rather, half of it), and remenance, height-wise.
Furthermore, airgap (when applicable, as in most inductors) reduces μeff, stretching out the B-H curve in such a way that the hysteresis loop is a smaller fraction of the total VARs being cycled; remenance is reduced, and Q factor increases.
