What alternative LEGO-compatible toys are avaliable?
"Communist LEGO" is a comprehensive list of all the LEGO imitators and clone brands. It is an interesting read: 
https://uploads.brickset.com/docs/clonebrands_v1.8.pdf

Keep in mind that, while clone brands may be cheaper at the beginning, they have minimal or no resale value, and thus can end up being more expensive than genuine LEGO. Because of LEGO's quality, reputation and popularity, you can always resell your collection once your kids outgrow them, and thus recoup the money you spent (as long as your kids kept them in reasonably good condition). Clones are hard to even give away, not to speak of selling.

Another thing to consider is safety. LEGO puts tremendous care and effort into making sure that their products are safe. This is not always the case with cheap imitators. Children often put toys in their mouths. You don't want to expose them to lead-based paints on printed items and leaching plastics.

Mega-Bloks and Kre-O should be safe as they are made by leading toy manufacturers with a long history. Their products are not the same quality as LEGO, so the resale value is not as good, but at least you know they won't poison your kids. It is best to stick with products with proven safety records rather than choosing toys just because they are cheap.

Also, you might consider the educational value of LEGO. It is not just a toy, but a system for building and learning. Math, science, coordination, engineering and creativity are all integral part of building with LEGO, and many schools and educational institutions use LEGO as part of their curriculum. So if you look at the whole package - quality, safety, and value - LEGO will still likely be your best option.

Based on your original post it appears that your children already have a good amount of LEGO, but they keep asking for more. I would suggest that you explore new activities with the LEGO they already have. LEGO provides many alternate building instructions, project suggestions, games, and lessons on their website, and there are also third-party websites, books and other materials to let your children explore the endless possibilities even a handful of LEGO bricks can provide.