Many papers I have seen describing transgenic rodent models (and presumably applicable to other model organisms) involve the knock-in, or modification to, a single gene, possibly two genes. With respect to recombineering techniques, what prevents targeting multiple genes in a single organism? For instance, if I wanted to simultaneously knock-in some genes and knock-out others within the same mouse, would I be forced to generate individually modified transgenic lines and then do some "fancy" breeding to generate the multiple-modified mice?
One reason is the low likelihood of success. Modifying a gene almost always involves a recombination event of plasmid DNA with a target site in the genome (and I say almost just because there may be some method that I don't know about, but all the ones I'm familiar with do). The likelihood of that decreases exponentially with the number of genes you're trying to modify. If you're trying to make several mutants of individual genes the likelihood of success decreases only linearly.
Another reason is having more knowledge and experimental power. You can learn little from a double mutant if you don't also have the individual mutants to compare. In fact, most reviewers would ask for individual mutant data if you've made a double mutant in your paper. This is especially true with flies and worms, as crosses take less time with them.
Also, the more mutant genes you have, the weaker the animal. Your mutants may not be viable at all with too many mutations.
You could introduce multiple transgenes into your embryonic cell line before injecting into the blastocyst. That way you don't have to do it individually one by one and then breed them all together, which can take a lot of resources. This is especially necessary when your genes are really close together (genetic linkage) and hence would not breed in a Mendelian ratio. Either way, each gene you try to modify usually takes at least a few months. I think CRISPR has made it faster, but still its a lot of gel blotting to make sure you generated the transgene.