What is the largest and/or heaviest object that could be sent to space? I know the space station is sent in pieces but I am looking with one launch.
Historically, the heaviest payloads that reached orbit were the Apollo missions; the third stage of the Saturn V booster plus the Apollo spacecraft combined exceeded 140 tons in low Earth orbit. The lower two stages alone could put over 120 tons in orbit; such a configuration was used to launch the Skylab space station in a single 77-ton piece.
Today, the highest-capacity operational launcher is the Falcon Heavy, allegedly capable of putting over 50 tons into orbit (though the figures are pretty unclear). Delta IV Heavy (28 tons) and Long March 5 (25 tons) are the 2nd and 3rd place finishers.
Unlike the largest historical launchers (Saturn V, Energia, and STS), which carried payload at least partially externally (if you count the STS orbiter as payload), the modern heavy lift launchers expect the payload to be entirely contained within a fairing, which sets limits on the physical size of the payload. Long March 5 and Falcon use a 5.2 meter diameter fairing, Delta IV a 5.1 meter fairing -- various different lengths are available.
I believe the plan is for SpaceX BFR to be able to take 150 tons into space:
There is no sensible limit to how much could be launched, just a mater of practicality.
Project Orion's aimed for it's interstellar craft to take 5500 tonnes to LEO, as a "conventional" launcher, Sea Dragon aimed at 550 tonnes.