So if a machine dissolves matter and puts it together again, couldn't you just repeat a stored information multiple times? That is reproduce a transporter signal over and over again?
Yes, this has actually once happened (to Riker, in "Second Chances").
After examining the Potemkin’s transport logs, Geordi comes up with an explanation—years earlier, as the transporter officer of the Potemkin struggled to beam up Riker through the distortion field, he tried the novel approach of splitting the transport beam. Part of the transport beam was reflected by the distortion field back to the planet, materializing an exact living replica of Riker. Unaware of what had happened, and assuming they had successfully recovered Riker, the crew of the Potemkin went on to their next assignment, leaving the duplicate trapped behind.
Transporter also once produced a merged Neelix-Tuvok hybrid on Voyager. Voyager had another attempt to create a person out of the transporter (Neelix's family) but that wasn't exactly a copy from the transported logs.