What is the largest possible size I can enlarge to with 35mm film?
There's an interesting and thorough paper explaining film resolution, granularity and print grain: http://cool.conservation-us.org/coolaic/sg/emg/library/pdf/vitale/2007-04-vitale-filmgrain_resolution.pdf
There is no clear answer to your question as it depends on multiple factors, namely:
The resolution depends on:
One can say that this is all unimportant, unless I suddenly want to make prints from, say, ORWO film shot by my grandfather in the seventies with Smena.
Also different methods of judging resolution lead to different results, magnification under loupe or microscope is different from enlargements which in turn is different from scanning and printing.
Finally, the viewing distance plays very important role in how big you can print. Kodak has published print grain indexes for their Portra line (page 7). It says that the grain of 35mm Portra 160 film is visible already in 4x6'' (10x15cm) prints, but without visual comparison or experience this does not mean a lot.
So the suggestion would be - test yourself, whatever numbers you get from here might not fulfill your expectations in the final print.
That all depends on the viewing distance, and, to a lesser degree the brand, age and speed of the film. As an example, ISO1600 B&W film will exhibit more grain than that from say, ISO 50 Velvia.
If you were shooting a large wall hanging to go in a restaurant or café, then you can get away with much larger enlargements than if you were wanting to hang this in an art gallery.
As a rule of thumb, I treat 35mm negs as the same as about a 24MP digital image, and would be happy to print an image for a gallery up to about A3 size; If it were more abstract, or somewhere where it sets the mood, rather than being the primary focus (so the restaurant scenario above), then I'd have no qualms with printing at a metre by just over half a metre (3' x 2') -- I've seen people go much larger than this, but you really are looking at moving to a digital process anyway then.
Film with a low ISO will typically enlarge better than high ISO film with lots of grain (unless, of course, you want that effect). Kodak produced a 40 x 60 foot print made from Kodachrome in Grand Central, but that's an extreme example.
Either way, the intended viewing distance matters a great deal. For normal viewing distance, with good film, you might get as large as 16" with good technique. For farther viewing distances, it could be much higher. Also, consider something like slide film which, when projected, is usually quite huge, but is also viewed from a distance.
Net effect, there's no simple answer to your question, it simply depends on several factors.
In short: you can enlarge it as much as you like.
Of course the larger you print, the more you'll see grain/noise, the more you'll notice the fine details of the picture are missing.
But the barrier for size is a purely aesthetical one, not a technical. Some will find a print in 3x5 too large, other will accept a 40x60 foot print as being great.
So it depends on the viewer of the picture if it is too large for 35mm, not on technical details.