Does there exist a non-electric bike that can go 60mph (95 km/h)? I recently ran into this article here where a woman bikes at about 184 mph (295 km/h). Now certainly there were some crazy ideal conditions here like an extra special bike and clothes, drafting behind a race car, and salt flats but that did get me thinking.

Would it be possible to make a bike that can go to decently high speeds without those conditions? Essentially a really good bike that has a LOT of gear levels so you can start with regular bike gear sizes and work your way up to something that could theoretically reach a very high speed.

Now 60mph is probably a bit realistic but what I am essentially wondering is what kinds of bikes are designed with the kind of gear system that would allow for the bike to both reach regular speeds and rather high speeds should the peddler have sufficient strength.

Now I would look to top races like the Tour De France to get an indication of what is possible here but I wonder if there are restrictions involved there that might be limiting that ability. So I suppose my question is: are there bikes that push the limits of human powered pedaling that can be started without driving behind a race-car?

Forgive me if this is a bit of a naieve question. I am fairly new to keeping track of what bikes are out there and what is possible here. I only ask because I am currently able to drive the highest gear on my bike (entry level 500$ bike) on a flat surface without much issue. I am trying to get a feel for what speeds may be possible should I find one with the "next gear up".
The problem with going 60 mph (assuming level ground, no wind, etc) is not gearing, which is an easy problem to solve. It's aerodynamics. Aerodynamic drag increases roughly with the square of your speed, and to get up to 60 mph on a conventional bike, you'd need to produce about 4000 W—which some sprinters can do for a few seconds.

So to go anything like that fast, you'd need to be in an aerodynamic HPV. The current hour record for a streamliner HPV is about 90 km, still short of 60 mph, although over sprint distances, HPVs have gotten up to 144 kph. 

However, these would be completely impractical for mass-start events like the Tour de France, so you can't easily extrapolate to what a mass-start HPV race would look like, or what speeds people would achieve, because the bikes would need to be significantly different. Some of these record-attempt HPVs have no windows at all, and rely on cameras and TV screens for outward visibility; the ones with windows have extremely limited visibility. Riders are taped in, and could not easily get out to change a flat, for example. 

Also, drafting plays an extremely important role in (conventional) bike racing. Because the benefits of drafting would be greatly reduced in HPV racing, those races would essentially turn into time trials.