The answer to this provocative question may surprise you. The answer also depends on exactly what you mean by "biggest?"
Relative to its body size, the barnacle wins the award for biggest penis hands down. When extended to its maximum length, the barnacle penis can be up to 42 times the length of its body! This makes sense for an animal that is attached to a hard substrate as a sexually mature adult. There are times a barnacle needs an extra long penis to be able to reach another mate since it can't walk over and ask a potential mate to dance.
For actual physical penis size, the blue whale has the biggest one in the world. Of course, the blue whale is also the biggest animal still alive on the planet, although there were some dinosaurs, now extinct, that were even bigger. The largest blue whale penises have been measured at 8-10 feet long which is about 2.4 meters! This whale produces about 200 pounds of semen per ejaculate.
Although most birds do not have a penis, the Argentine Lake Duck has the biggest one of those birds that do. It is about 17 inches in length. This duck's penis is also the largest penis relative to body size for the vertebrates (animals with a backbone). Of the primates, the human male has the biggest penis (and the longest stamina on average). Another penis of notable size is that of the deep sea squid which is well over 2 meters long!

