I guess a better way to ask the question would be, Are there compounds that exist in biology that humans don't have the capacity to produce 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 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.
I work in a natural products synthesis lab, so your question is exactly about what I do every day.
There is the term ‘paper chemistry’, which describes reactions that one has drawn on a paper (a rather easy task in a chemist’s lab life) but which one hasn’t actually performed yet. The vast majority of a Ph.D. student’s work is paper chemistry: Thinking of how to synthesise a molecule, attempting the reactions, finding out that something unexpected happens, rationalising it, facepalming for the silly idea one had and moving on to the next paper chemical reaction. With paper chemistry, everything is possible.
In reality, it usually takes a Ph.D. thesis for a molecule of average complexity. Some colleagues are lucky and get it done in a year because everything worked, some are unlucky and don’t get through with their project in four and a half years because nothing worked. Or it worked, but only in $10\,\%$ yield. Or it worked but the protective group fell off and we need to reistall it; all those little nastinesses of everyday lab work.
Nature manufactures its biomolecules, both primary and secondary metabolites, extremely efficiently; usually with yields close to $100\,\%$ and in a very atom-economic way of synthesis. Lab chemistry usually deals with yields of less than $95\,\%$, separation of undesired side products, and uses bulky protective groups. Many natural product university classes show two pictures of how nature does it (assembly-line style) and how chemists do it (a smith with a hammer) just to emphasise these differences.
The reason for this efficiency difference is 4.5 billion years of optimising that nature has gone through, while mankind has had only about 150 years time to optimise its chemistry. Maybe our efficiency will catch up one day. Maybe not.
On the long run, I don’t believe that there is a molecule out there that cannot be synthesised in vitro. On the other hand, there are examples like maitotoxin (19 years as of now) that show just how long it can take for just a single molecule. I believe I read about a molecule whose total synthesis took 24 years in Classics in Total Synthesis III — that’s the linear equivalent of 8 Ph.D. theses! So it is probably up to mankind’s endurance whether the extremely complex molecules will be synthesised or not.
Hard question, but I suspect the answer is yes. The original definition of organic molecule was one that could only be manufactured in an organism, which turned out to be false. It may be true that there are molecules we do not yet know how to synthesize, but that does not preclude our figuring it out in the future! We managed to synthesize vitamin B12, for example, which was a major feat. So there may well be compounds that we cannot synthesize today, but that is not good indicator of what we could do in the future.