Can beef stew be reheated several times?

There is some very useful information for you in this thread.

The direct answer is, every time you re-heat and cool the entire pot, you're passing through the "danger zone" (40-140 °F or 4-60 °C) where bacteria will continue growing. Even if you manage to kill all of them, this is a problem because:

Some bacteria leave behind harmful protein toxins that cannot be "killed" (denatured) by cooking. Cooking food is only effective against live organisms, not their toxic waste products.

So yes, there is a very real risk. To reduce this risk as much as possible, you'll want to follow the alternate method you describe of only re-heating the amount that you want to eat at any given time.

That's also desirable because:

• If you bring the entire pot back up to boiling each time, you're continuing to cook it. Besides the risk of contamination, that will also affect the quality and palatability of the stew because it will be drastically overcooked after just a couple repetitions.
• It's a lot more efficient to apply just the heat needed to re-heat a small portion than it is to re-heat a larger amount that you're just going to let cool back down anyway.

Also, keeping your stew in the pot may be undesirable, especially if it has acidic ingredients like tomatoes, which can react with a metal pot (particularly aluminum) and give your stew a faintly metallic taste.

Do yourself a favor and don't try to to take this shortcut. It's a bad idea for a number of reasons. Pack the stew into individual servings instead and keep them refrigerated until you're ready to eat them.