44 Mag is also all over the place in terms of power and quality of projectiles. Some lower power cowboy lead ammo is not surprisingly cheaper than some full power jacketed soft point or hollow point ammo. What is it you are wanting to do with the gun? And what does "inexpensive" mean to you?
You mentioned looking online, so I'm sure you know about gunbot, ammoseek, slickguns, etc. In my experience, full power jacketed stuff is about 65-75 cents a round. To shoot cheaper than that, it will be lead/cowboy stuff or commercial reloads. Or you need to consider rolling your own.
FWIW, for a jacketed bullet:
HSM .44 Mag 240 Grain FMJ-PFP 50 Rounds - 193440, .44 Remington Magnum Ammo at Sportsman's Guide
They have free shipping for labor day, the code is in the banner on the website. I've heard this ammo is sort of "soft" in terms of power, but have never used it.