Can all compounds that plants and animals produce be synthesized in a lab?
I suspect, in its current form, this question is largely opinion based. Currently, as it stands in September 2015, there are definitely limitations. But never say never. 

On the small molecule scale, Maitotoxin is a famous example of a molecule that has yet to been synthesised in the lab. But perhaps only for a matter of time (or money)

3D printing and automated synthesis is expanding the scope for small molecule synthesis, giving more flexibility to synthesising recalcitrant targets.    

On the larger scale, human DNA contains up to about 300 million base pairs. Currently, the largest DNA synthesised in the laboratory is about 600,000 base pairs, so still a way off. However, with increasing capacity in areas like artificial gene synthesis, complete replication of entire DNA is not unforeseeable. One day.