I currently use Photoshop CS3 and really have no desire to upgrade at this time (long story don't ask) I am thinking about purchasing Lightroom 6 and I see that the manual for LR6 states that it will work with CS3. Has anyone had any experience using this combination? Thank you.
No reason it shouldn't work.
The "Edit in Photoshop"-function will be limited, but it'll work. Full functionality requires the ACR plugin in Photoshop to be at equivalent version level. You also lose some other "Open As..."-functions into Photoshop that likewise require matching versions.
But the basic functionality - editing in Lightroom and sending the finished result to Photoshop - will work.
Disclaimer - like Jitendra I haven't actually tried it.