SpaceX has predicted that Starship will be able to lift up to 150 tons to LEO. Could some of the payload that it will carry have a third stage? If so could that save another flight of Starship since then Starship would not need to get refueled in LEO.
Technically, yes. But not really.
From the Starship user manual, the payload bay of Starship is 8 meters wide and 17.24 meters tall, with an option for an extended 22-meter fairing. In terms of length, this is pretty similar to the Space Shuttle's 18.3-meter long, 4.6-meter wide cargo bay.
I came up with probably the only useful application of an extra stage on Starship - Centaur. This is because other stages will not provide a useful amount of delta-v for deep-space payloads. While other stages like Delta-K can be useful due to storability, it is small enough to fit as a secondary payload (rideshare capability?!), while the Centaur could be fitted to boost a payload to even higher-energy destinations.
The main plus about this is that Starship can carry a Centaur III (23 tons, 3m diameter, 12.68m length) and some more into GTO with extra delta-v to spare, so maybe a Centaur with payload could be lifted to TLI with a Starship without refueling.
However, there are a few drawbacks to this option. First of all, Centaur and Starship belong to two different companies, ULA and SpaceX, respectively, and commercial cooperation is difficult when there is no mediator (like NASA). Aside from business regards, it would be difficult to fuel the Centaur while it is in the payload fairing, so fuel boiloff would be a major problem should Centaur be fueled prior to encapsulation (safety hazard!). However, if a Starship-Centaur is needed, SpaceX probably should just build a variant of the Starship specifically designed to accommodate a Centaur (or of course, some other cryogenic stage).
Overall, there isn't really a point in putting an extra stage on Starship as it could be refueled (expending the first Starship, probably). If SpaceX does choose to add a third stage, it most likely would be either a collaboration with ULA (Centaur) or the company developing a high-energy upper-stage similar to the Centaur, but development costs will be very high so it is not economically feasible.
Starship is targeting 100-150 t of payload and has a sizable payload volume, so it certainly could carry a third stage. However, this would come at a cost of part of that payload mass and volume, and of adding a large piece of expendable hardware.
Starship is designed to take advantage of orbital propellant transfer. One tanker launch will greatly extend its range of accessible orbits, without expending any hardware or requiring any new stages to be developed and provisions made for safely loading and unloading whatever propellant they need. It's not intended to be some unusual operation that they will take exceptional measures to avoid.
If a third stage is used, it likely won't be to save a tanker flight, but to save a Starship that would otherwise be difficult to recover, for example on a far interplanetary launch. Even in this case, though, you have to compare the cost and limitations of developing a third stage that fits within the cargo hold and competes with the payload for mass and volume, and using an expendable Starship which would just be a simplified version of what's already in serial production.
Pretty much any GEO mission launched on Starship is going to need something to get to GEO. Thus the third stage will come with the payload at a minimum. (Short of some kind of refueling mission and then using Starship to boost to a GEO transfer orbit. Which is probably not very likely).
So in that technical sense, yes. What you more likely intended was a more 'formal' third stage and that is yet to see. But I suspect most will be carried inside the upper stage, released and used after release.
Perhaps the Starship (modified) could be used as a second stage on the Artemis SLS.
In it's current form, Artemis rolls to the launch pad straight out of the 1980's "Like a Virgin" and spends money "Like a Material Girl".
Future iterations could have a recoverable first and second stage.
Combining the Artemis and Starship programs could bring out the best in both designs.
The strap on solid boosters would reduce the number of Raptors needed on the first stage. Amazingly, the Shuttle era RS-25 and the Raptor 2 engines have very similar thrust.
A recoverable Booster with fewer Raptors could serve as the SLS first stage, with the second Starship derivative stage flying to a powered landing down range as any type of third stage proceeded into orbit.
This would lower costs per launch and give the SLS tremendous heavy lifting flexibility, giving it a greater chance of paying for itself, or even generating a profit.