Is the number of rays projected by a source of light finite?

A ray of light is just an approximation, that treats light as a linearly propagating (ignoring the wave-like behavior of light). It is applicable in certain situations, such as ray optics. Under the right experimental situations, treating light as a ray will not be appropriate.

So, a source of light giving out infinite rays is a convenient approximation that neglects the fact that light is made of photons and so on. Also, it's not meaningful to talk about each ray containing so many photons. If you know the power of the light and its wavelength, you can calculate the number of photons emitted per second (their rate of emission). In that case, you're not using the ray picture!

A light ray isn't an object, it's just a direction. We define the ray to be the direction in which the light is propagating. So while there are an infinite number of possible rays this just means there an infinite number of possible directions for the light to propagate in. You shouldn't think of a ray as an object containing one or more photons.

Also, it's not correct to think of a beam of light as being made up from photons. A photon is the unit of interaction of light with something. For example, in the photoelectric effect the light wave interacts with the metal in units of photons, with each photon, i.e. each interaction, ejecting one photoelectron.

A good definition for Ray of light is provided by Wiki. It is a theoretical term ideally used to mention the propagation of light in Ray optics (simply, it's an assumption of a path that light may take along its direction). You could draw infinite number of rays from a point source of light. We require rays (at least wave-fronts) for drawing diagrams of reflection, refraction, etc.

According to Wiki,

It is a line or curve perpendicular to the wavefront of light.

If we take Huygens principle (Every point on a given wavefront may be considered as a source of secondary wavelet which spread out in the medium at $c$ and the new wavefront is the forward envelope of secondary wavelet at that instant) into account, we could do a lot of tactics here.

The number of photons in a ray of light at any given period of time (i.e. the rate of emission) is finite. But, that number of photons is also based on our assumption (how we've chosen the ray). Also, the source is always emitting photons until it's switched off...

Photon is just the quantum of electromagnetic radiation or the carrier of EM energy (or force) having zero rest mass, exhibits wave-particle duality and which has the anti-particle only as itself. While calculating the frequency (or wavelength) of the light (or photon), you aren't considering a ray of light 'cause you don't require the use of it.

I can't still understand how the energy became infinity? It becomes infinite only when the frequency is $infty$ or it's an infinitely energetic source (which is too ideal here).

I've had the same question but let's look at it differently. As I a one one dimensional line: Let's say we have an illuminated line with a finite length. We know that the photon is a particle hence it has diemensions. Now let's suppose this lign is illuminated by photons. So the photons are arranged to cover the whole length of the line. Now let's suppose this line has a length L( finite) And the diameter of a photon is d( finite) then we have L/l photons ( finite). Now we can apply this to a 2 dimensional plane with a surface S if we suppose ds the surface of.a crossection of a photon then this surface is illuminated by S/ds photons ( finite ) . This way we don't have this confusion of infinite energy since a light ray is nothing physical. It's just a concept created to explain certain situations