What are the most basic parts of a bicycle which cannot be upgraded over a period of time?

Any and every part of a bicycle can be replaced, including the frame. (Many people would probably regard a different frame as a different bicycle though.)

You ask whether there are components that 'cannot be upgraded'. I think you are misusing that word. Anything can be upgraded, i.e. replaced with a equivalent but better quality version.

I think what you mean is that you cannot change the basic design of the bicycle. As you note in one of your examples frames are designed for a specific wheel rim diameter. You cannot put in a larger (or smaller) wheel because it will not fit or you will mess up the steering geometry.

By the way, your assertions about aluminium frames being stiffer than steel, larger wheels holding more speed and larger diameter tires having a larger ground contact patch are all untrue.

The bicycle's serial number is usually found under the bottom bracket. if you regard the serial number as equivalent to the VIN number on a car, then changing the frame makes it a new bike.

The problem I see with this line of questioning is that it assumes the existence of some ideal perfect bike that has every conceivable upgrade on it. Then you are measuring a bike you're considering buying against that ideal with the idea that you could upgrade part-by-part until you have the ideal. That just doesn't exist because bikes are built for different purposes (road, cross country, downhill, etc) and have different design and cost considerations. Also, the ideal changes over time as new innovations come on the market. Some of those innovations are backwards compatible and others are not. As others have said, ANY part CAN be upgraded to a better quality one. However, there are compatibility issues that will always come up (wheel size, bottom bracket and headset standards, etc). You can never have a bike that allows for ALL possible upgrades.

I'm just going to point out some minor things the other answers have missed.

The original question correctly pointed that you may be unable to change the wheel size, and you certainly can't go from small to big wheels because there won't be room. You will also change the handling in ways you didn't expect. Many gravel bikes are designed to accept 700c and 650B wheels (these being equivalent to 29" and 27.5" in the MTB world). With 650B wheels, we would run larger tires. If your bike isn't designed to take 650B wheels with larger tires, you are likely but not guaranteed to run into clearance issues. You would certainly not be able to do this on a road bike. I have a Parlee Chebacco, which is designed for 700c alone, but I have heard of one person fitting 650B wheels with relatively narrow tires (which, in my view, defeats the point).

Related to the above, frames all have a maximum tire size they can fit. Many older road bikes from the 1990s and 2000s may have been designed for 23mm to 25mm tires. Road bikes are trending towards 28mm tires right now, because they are more comfortable and no slower than narrower tires. However, older bikes may be unable to upgrade to these. Similar issues might exist for some gravel bikes; mine is designed to accept 40mm tires, but the trend is towards 45mm or wider.

You can't change from rim brakes to disc brakes (although a handful of frames may have been designed to accept both rim and disc brakes). You can change the fork, which gets you one disc brake, but the tubing at the rear of the frame would not have been designed to accept the stresses of disc braking. Thus, even if you were able to drill disc brake mounts, your frame would probably be under more stress than it was designed for, and it may be too flexy, and it might even fail. (NB: drilling the frame also creates stress risers, which can lead to later failures - and under the stresses of disc braking, they very likely would do this.)

Internal cable routing is currently the dominant trend for drop bar bikes. Bikes designed for external routing can't generally be upgraded to internal routing. One possible exception is that if you're talking about electronic shifting, it's possible to drill steel and titanium bikes for internal electronic wiring. As noted earlier, drilling poses some risk due to stress risers, so this should be done by a professional, and it's at the rider's risk. (Of course, there's also wireless electronic shifting.)

In general, the standards for mounting components to frames sometimes change. On road bikes, we recently went from quick releases to thru axles. On mountain bikes, thru axles have been dominant for longer, but MTB frames are going to Boost spacing, which has wider spacing for both front and rear hubs. Head tube diameters are another example of a change like this (my custom bike is designed around a 1" steerer, most bikes in that era used 1-1/8" straight steerers, and even wider and sometimes tapered steerers are being used now). This should be distinguished from component standards changing (e.g. freehub spline design, like Shimano moving from Hyperglide to microspline, or Campagnolo changing its freehub design from 8s to 9s); with those, you just have to buy more stuff to make an upgrade.

The question is also inseparable from the issue of how economical it is to upgrade parts. Because bike manufacturers buy components in bulk, they get much better deals than consumers do. Past a certain point, it doesn't make economic sense to upgrade a bike, compared to selling it and buying a new one. Naturally, this may be irrelevant for frames with a lot of sentimental value, e.g. custom frames. Also, as pointed out, it is physically possible to change everything out barring incompatibilities above, including the frame. If you do the latter, you do raise an ontological question of whether you have an entirely new bike or not (I suspect most people would say yes, as alluded to in other answers).

I think you'll find that the second most difficult component of a bicycle to change is a wheel hub (unless changing the entire wheel obviously). This means if you are interested in for example dynamo hub lighting system, you should try to find it as stock hub in an existing bicycle. The most difficult component of a bicycle to change is the frame. Changing a frame might affect the possibility of using certain brake types (short reach caliper brake vs long reach caliper brake vs cantilever/V-brake vs different mounts of disc brakes). Also different frames have different axle mounts (normal axle and what over-locknut-distance? vs thru axle and if thru axle then what diameter and what length?). The front derailleur mount might also differ. Very likely you cannot use the same seatpost if you buy a new frame. Supported headsets also differ between frames. Bottom bracket mounts also vary.

Example #1: There are two cycles - Cycle A's frame is made of Steel and Cycle B's frame is made of Aluminium. We know that frames made of Aluminum are generally stiffer than steel which saves few fractions of a second count [citation needed]. Let's say I bought a Steel Framed Bicycle and want to make it more track-oriented and save a few microseconds off. I cannot have a stiffer frame because I cannot upgrade the material of the frame. If only someone would have told me that I should have invested more on a bicycle which had an Aluminium frame because I cannot change it.

You seem to assume lack of stiffness equals to power loss. Lack of stiffness doesn't equal to power loss.

Heavy touring cyclists hauling their heavy loads with heavy steel framed bicycles don't complain about lack of stiffness. In contrast, a typical racer plus racing bike weighs less than half than the touring rig plus heavy rider plus load.

Example #2: I want to install a larger size of rims from 26" to 27.5" because Larger wheels hold more speed than smaller wheels due to rotational inertia and more grip because the larger wheel has more rubber touching the ground.

I would say it's the 70 kg rider holding speed, not the 2 kilograms of tire, tube and rim at the wheel perimeters. Yes, rotating mass near the edge of the wheel is counted twice but it's still far less than the rider.

Also do note that rotating mass is counted twice regardless of the wheel circumference. You seem to assume larger wheels would count the rotating mass with a larger factor which isn't true. Larger wheels rotate at slower angular rate which cancels their larger moment of inertia.

Also, most people are interested in reducing the rotating mass because part of the rotational energy goes to braking. So it's more bad to have lots of rotating mass than it's to have little rotating mass.

Also wheel size doesn't enter into the friction coefficient.

Example #3: I want to upgrade the derailer from 24 to 27. I know it's possible, thanks to Bicycle Stack Exchange, I got good guidance for the upgrade and got lucky with the fact that my cycle allowed me to upgrade the derailer. If my cycle didn't have space for a 27-speed crank then it would not have been possible for me to upgrade. Let's say I bought a 21-speed bicycle because it's cheaper then I mostly wouldn't have had any space for a 27-speed crank or even a 24. I would have been stuck with my 21-speed until I buy a new cycle.

The rear derailleur isn't speed specific in the speeds of interest.

(Of course, some more recent designs have different cable pull ratios but even then it isn't only the speed count that tells the cable pull ratio. For example, Tiagra 10-speed 4700 uses the same cable pull ratio as Shimano 11-speed road groupsets. The older 10-speed groupsets have a different cable pull ratio. But this doesn't interest you as you are interested in 8 and 9 speed components which have the same cable pull ratio apart from old Dura-Ace 8-speed.)

When you speak about "upgraded over time" I think you mostly mean "ages well". Every time I buy a bicycle I want to make sure that it stays up to date for as much time as possible. And I try to sell it before the technology behind it gets out of date. The scope narrows down to the frame, pretty much, but components are also important, as they can make upgrade financially unreasonable.

Practical tips:

• Avoid proprietary stuff, old or unique standards, and other unique essential components that may leave you with a frame that has no spare parts, or nobody wants it due to bad reputation of some parts. Examples: older Lefty forks, custom seatposts, frame materials, custom headsets, Trek's BB90 that they just abandoned, older axle widths, brake types, and even "not cool" geometry and "old" looks. There are many cheap steel frames on eBay, but if you buy them you'll find that there are no compatible components available anymore.
• Don't get into the components trap. Some component setups may lead to a very expensive upgrades if/when needed. There could be no smooth way to upgrade to the latest trends, but rather a cliff. For example, older Shimano DI2 have no compatibility with newer versions, so potentially you'll need to replace every single thing. Sram AXS release made all the older Sram components totally unsexy, I see people having problems selling bikes with Sram Red, and there is no way to upgrade without changing EVERYTHING, up to a freehub body.

Real life practical story:

• I decided to sell my expensive rim brake carbon superbike with 2x11 Ultegra 6800 before it's too late and price goes down. Rim brakes are losing demand drastically on the 2nd hand market, plus Shimano is about to release 12 speed for road, so while Ultegra 6800 doesn't look old against newer R8000 Ultegra today, that bike would look truly ancient when 2x12 comes in, even being 3 years old flagship bike!