Paired end sequencing with R1 and R2 of different length. Possible?
As indicated in the comments, yes, you can definitely tell standard Illumina sequencers to sequence mates in a pair to different lengths. This is quite common in single-cell RNA-seq, where R1 is often just a cell barcode and an UMI, so sequencing it to 150 bases would be a huge waste or reagent. One caveat to this is that you can't buy reagents sized for these custom length, just things like 150 base PE or other standards. So unless you end up saving and pooling reagents after such runs you end up paying the same for projects with one shorter read than if you'd just directly sequenced everything to the same length. Given that sequencer runs only take a day or two at the moment, this also doesn't result in that immense of a time savings.