Can a computer run without an operating system if it has a good application?

Answer 1

Yes, the earliest computers built did not have operating systems (the idea of an operating system had not been thought of) so individual programs were loaded and run one at a time. The first computer I used (in 1973) had no operating system, programs punched on decks of cards were loaded and run one at a time.

The first microcomputers sold did not have enough memory to load an operating system. When Bill Gates wrote the original version of Microsoft BASIC for the MITS Altair 8800 the standard 8800 came with only 256 bytes of RAM and Microsoft BASIC could not run with less than 4096 bytes of RAM (it was already very expensive to expand the RAM to that size), so Microsoft BASIC was written to run without an operating system.

There are also embedded system computers that are dedicated to one purpose and usually run one preloaded program supplied by the manufacturer and do not allow other programs to be loaded. While some embedded system computers do use an operating system, it is also common for the preloaded program to be written to directly control all the hardware directly without the need for an operating system.

Answer 2

Yes; it's just easier to design programs that work with an operating system, since the operating system takes charge of many important details, which you then won't need to program any more.Also, such a program would basically need to take full control of the computer, so it won't cooperate well with other programs - such cooperation is also achieved by an operating system.

One example of such a program would be type-1 hypervisors; i.e., certain types of virtual machines.