Is it possible to create a file that never completes its download process?

Yes it is possible. You just need to use the chunked transfer encoding. https://en.wikipedia.org/wiki/Chunked_transfer_encoding

Depending on your server's configuration, you might be able to simply create a CGI script that writes and flushes stdout in an infinite loop.

It does not seem to work on Lighttpd which I believe buffers the entire output from the CGI script before sending it to the client. It might work on other webservers though.

Example:

HTTP/1.1 200 OK\r\n
Transfer-Encoding: chunked\r\n
Content-Type: text/plain\r\n
\r\n
1e\r\n
Uh-oh, this will never stop.\n
1e\r\n
Uh-oh, this will never stop.\n

followed by an infinite repetition of "1e\r\nUh-oh, this will never stop.\n"

Yes, just stream /dev/urandom to the client. First, maybe you'll need to fake the file header, so that client thinks it's downloading the stuff it requested, and after that just stream random junk.

An idea on how to do this in Python:

with open("/dev/random", 'rb') as f:
    print repr(f.read(10))

If the output is dynamically generated by the server software, it is possible to create a stream that keeps going on until you break the connection. However if you literally want a file it cannot be infinite in size.

However if your file system supports sparse files you can create a file that is larger than the storage media and that way produce a file which would take such a long time to download that it isn't feasible to download it all.

The maximum file size differs between file systems. On ext4 the limit is 16TB. On tmpfs the limit is 8EB. Here is a couple of examples on how such files could be created:

dd if=/dev/null of=/dev/shm/sparse bs=1 seek=7E
dd if=/dev/null of=/tmp/sparse bs=1 seek=15T

Beware when putting such files on a webserver. If the server software you are using doesn't throttle the bandwidth it is possible for a malicious client to overload your network.