It is probably impossible to get cells (any cells) to reproduce at 10 times in a second. Very simple bacterial cells in optimum conditions for growth, for example, have a doubling time of ~ 20 min. Cellular reproduction requires a considerable amount of biological material to be synthesized (e.g. DNA, proteins, lipids, carbohydrates). The rates of the chemical reactions and the rates it takes these compounds to be transported throughout the cell would place a fundamental limit on the doubling time of an cell. For example, the rate at which DNA polymerase enzymes can synthesize DNA is somewhere in the range of 10-100 nucleotides per second, so it would be difficult for these enzymes to synthesize even a minimal genome in a fraction of a second. Even if these enzymes were made to be "perfect catalysts" (the reaction rate is limited only by the rate of diffusion of the reactants), I don't think they could support a reproduction rate as high as 10 times per second. My guess as to the upper limit would be on the order of the rates seen in bacteria: tens of minutes. Perhaps one day we could design simple cell-like systems with doubling times on the order of minutes.
It is possible, however, to engineer human cells in ways that they can't stop reproducing. This is, essentially, what cancer cells are: cell that have accumulated mutations that allow them to replicate uncontrollably and indefinitely. A good discussion of factors required for cells to become "immortalized" is given in the following review by Hanahan and Weinberg, "The Hallmarks of Cancer" (http://www.weizmann.ac.il/home/fedomany/Bioinfo05/lecture6_Hanahan.pdf ).