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I can think of one particularly famous SpaceX 2nd stage in heliocentric orbit that even has its own website: https://www.whereisroadster.com/ and I know one was used to send DSCOVR towards Sun-Earth L1, though I'm not sure how it got there:

But it's the SpaceNews.com article about an upcoming launch NASA awards DART launch contract to SpaceX got me thinking, how many Falcon 9 2nd stages are "out there" beyond Earth orbit?

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    $\begingroup$ IIRC, the second stage from the TESS launch was sent into a heliocentric orbit $\endgroup$
    – Jack
    Apr 21, 2019 at 7:10

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According to this list on Wikipedia nine SpaceX second stages were launched into heliocentric orbits. No other SpaceX rocket went beyond a geostationary transfer orbit (GTO). In launches to GTO the second stage is typically left in a decaying elliptical low-Earth orbit, until it re-enters the atmosphere. (see e.g. here, here or here)

  • DSCOVR was sent to the Sun-Earth Lagrange point L1. Without station keeping this Falcon 9 second stage will have left this point and is now somewhere on the edge between heliocentric orbit and Earth's hill sphere.

  • Elon Musk's Tesla Roadster and the second stage that launched it are on an elliptical heliocentric orbit that crosses Mars' orbit.

After a long-duration coast, the Falcon Heavy’s upper stage engine reignited to give the Tesla and its mannequin passenger enough speed to break away from Earth’s gravitational grip, sending the car and its rocket stage into an orbit centered on the sun that will travel as far away as the asteroid belt. https://spaceflightnow.com/2018/02/07/spacex-debuts-worlds-most-powerful-rocket-sends-tesla-toward-the-asteroid-belt/

  • TESS was launched onto a lunar flyby trajectory, after which the Falcon 9 second stages was placed in a heliocentric orbit:

For the Falcon 9 second stage, the day was not done at TESS separation as SpaceX planned to dispose the upper stage into a heliocentric orbit by firing its engine a third time to escape Earth’s gravitational influence.

https://spaceflight101.com/tess/falcon-9-launches-tess-planet-hunter/

  • DART was sent on a collision course with an asteroid, which of course orbits around the sun:

But because DART was launched as a dedicated Falcon 9 mission, the payload along with Falcon 9's second stage was placed directly on an Earth escape trajectory and into heliocentric orbit when the second stage reignited for a second engine startup or escape burn.

Because Danuri was launched as a dedicated Falcon 9 mission, the payload along with Falcon 9's second stage was placed directly on an Earth escape trajectory and into heliocentric orbit when the second stage reignited for a second engine startup or escape burn. https://en.wikipedia.org/wiki/Danuri

At T+plus 40 minutes, with the rocket soaring over Africa, the upper stage reignited for a nearly minute-long firing to propel the payloads on a trajectory to escape the grip of Earth’s gravity and head into deep space. https://spaceflightnow.com/2022/12/11/falcon-9-ispace-mission-1-live-coverage/

  • The Euclid telescope was launched to the Sun-Earth L2 point. So like DSCOVR, the upper stage will be between heliocentric orbit and Earth orbit.

  • The Psyche spacecraft was launched into a heliocentric orbit intercepting the asteroid 16 Psyche.

The single engine powering the rocket’s second stage then took over the climb to space. After a 45-minute coast, the engine fired a second time, putting the vehicle on the required Earth-escape trajectory. https://spaceflightnow.com/2023/10/13/falcon-heavy-launches-nasas-psyche-asteroid-probe/

The following eleven launches at heliocentric velocities are planned:
IM-2 Nova-C lunar lander (TLI), IM-3 Nova-C lunar lander (TLI), Blue Ghost M1 (TLI), Europa Clipper (Heliocentric), Hera with Juventas and Milani (Heliocentric), Astrobotic Griffin lunar lander (TLI), ispace 2nd lunar lander (TLI), Interstellar Mapping and Acceleration Probe (IMAP) (Sun–Earth L1), PEE&HALO Lunar gateway station modules (TLI), Astrobotic's Third Lunar Lander (TLI), Nancy Grace Roman Space Telescope (Sun–Earth L2)

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    $\begingroup$ The linked Wikipedia page has been changed in year plus since you wrote this. The Falcon 9 upper stage that put DSCOVR into its Sun-Earth L1 pseudo orbit apparently never did make it to heliocentric orbit. That upper stage instead has been orbiting a bit chaotically within the Earth's Hill sphere, and will crash into the Moon on March 4, 2022. $\endgroup$ Feb 3, 2022 at 12:43
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    $\begingroup$ @DavidHammen Thanks for the info. I will update the answer. Feel free to edit the answer the next time. $\endgroup$
    – KarlKastor
    Feb 3, 2022 at 22:04
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    $\begingroup$ @DavidHammen Apparently this was a misidentification and it was actually a Chinese rocket stage hitting the moon. So I rolled the answer back to the original version. $\endgroup$
    – KarlKastor
    Feb 13, 2022 at 13:35
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    $\begingroup$ Thank you for keeping your answer up to date! I suppose you'll have to stop by again in March :-) $\endgroup$
    – uhoh
    Feb 13, 2022 at 14:48
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    $\begingroup$ uhoh! Astronomers now say the rocket about to strike the Moon is not a Falcon 9 $\endgroup$
    – uhoh
    Feb 13, 2022 at 23:38
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I'll supplement @KarlKastor's "4" answer and highlight some future missions at the end that will up the total:

4

Past missions

  1. According to Wikipedia, the Triana (also known as DSCOVR or GoreSat) was launched by the NOAA (National Oceanic and Atmospheric Administration) on February 11 of 2015. It's purpose was to measure the weather and climate in space. It also was used as an Earth observation satellite. It was sent to the Lagrange Point, L1. A Lagrange Point is a pint in space where the gravity of 2 bodies, in this case the Sun and the Earth balance each other out. The second stage is now somewhere between the Earth's and Sun's spheres of influence.
  2. The TESS (Transiting Exoplanet Survey Satellite) was launched on April 18 of 2018. It's purpose was to discover new exoplanets around the brightest stars for a 2 years period. It would accomplish this by using the transit method. The transit method involves noticing the slight dimming of the host star when the planet passes around it, indicating its existence. While the mission itself was launched into lunar flyby, the second stage is in a heliocentric orbit.
  3. The DART (Double Asteroid Redirection Test) was launched on November 24 of 2021 by NASA. It's purpose was to test a method of deflecting asteroids or other dangerous NEOs (Near Earth Objects). In approximately 4 months, the craft will intentionally crash into the asteroid Didymos to test this. Since the asteroid, of course orbits around the Sun, that is where it's second stage is.
  4. An, of course, how could I forget about Elon Musk's Tesla Roadster. It was an electric car made by his company TESLA. It was launched as a dummy payload for the Falcon Heavy's maiden flight. It, and it's second stage, orbit around the Sun.

Future missions

Of course, there will more. For example, the Psyche mission is planned to explore the origins of planetary cores. JANUS may join it to explore asteroids with its dual spacecraft. Also, while the Europa Clipper was originally planned to be launched on the NASA SLS (Space Launch System) rocket, it has now been moved to a Falcon Heavy rocket. It going to be a Jupiter orbiter mission, which will flyby various of Jupiter's moons. Namely, it will fly by Europa no less than 44 times. So, if you count future missions, then 7.

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