Why is ATP the most prevalent form of chemical energy storage and utilization in most cells?
I really like this question as it is such a fundamental underpinning of all life on the planet, yet there is such sparsity of actual information on its origins and why selection rewarded ATP use over anything else. Here I am talking generally since no specific studies exist in ATP vs other candidates.
A lot of the below information is taken from a relatively old article mentioned in the comments by TomD that discusses: "Why nature chose phosphates." by Westheimer, 1987. The article is very influential and has been cited over a thousand times since publication. I've decided to make a revision to my original answer upon discovering another article that came out in the same year that this question was asked "Why nature really chose phosphate." by Kamerlin et al., 2013
Some of the below arguments are more convincing than others.
ATP is an efficient and relatively simply biosynthesised molecule that can fulfil multiple biochemical roles. Cells do have alternative energy carriers, some with more specialised roles, however ATP is ubiquitous throughout our cells and intercellular spaces. There isn't a wealth of resources explaining why ATP is any better than other compounds, however there is plenty of reasons why the phosphates are required.
Why not the alternatives?
Citric acids and their derivatives are a good candidate, with deductible groups and high bioavailability but they simply don't give enough energy to stabilize genetic material.
Another tribasic candidate is arsenic acid. This is a fundamentally toxic compound though, which isn't particularly great for living things.
There are other phosphates too, and they are used in many organisms. But they have specific functions, and not used as the general energy carrier. For example creatine triphosphate provides a high energy phosphoanhydride bond, that is often used to quickly and anaerobically regenerate ATP, useful during high rate muscle activity for contraction.
GTP is structurally very similar to ATP. GTPases are used more to initiate cellular signalling pathways. It is is sometimes used as an energy source. This is a good example of an alternative energy carrier.
Over the years, many proteins have specialised with a specific shape, and this chance is the primary reason behind ATP over GTP. In other words, the choice of ATP over GTP is primarily down to cellular preference of molecular shape. One of them had to emerge as being more widely used, and it was ATP that 'won'.
Efficiency and simplicity.
The reaction was once thought to be a relatively simple nucleophilic displacement. From the 2013 paper:
Traditionally one will be taught that ATP is such a chemically efficient way of storing and transporting energy. This is due to the ATP->ADP & Pi hydrolysis reaction. The phosphate groups in ATP are full of negative charges and these are repelling one another. This means that the third phosphate is a great leaving group and breaking the phosphoanhydride bond is a favorable reaction. ...
...But the story is a lot more complicated than that. The above explanation isn't really satisfying because those same negative charge forces are repulsive of the nucleophile that is attempting to complete ATP->ADP & Pi. A more comprehensive explanation would go along the lines of: although a negative charge repulsion exists between the nucleophile of the protein and the phosphate, that high energy barrier can be overcome by electrostatic manipulation. This allows an "on-off switch" for the hydrolytic reaction by tweaking the electrostatic environment. This is another great regulatory tool that the phosphates provide. This regulatory feature is important for signal and metabolic/catabolic cascades.
When it comes to 'rebinding' the Pi to ADP, it is fairly easy since ADP seldom covalently binds to anything, which would require a lot of energy to recover the ADP. This also helps the bioavailability of free ADP to ATP synthase, an incredibly efficient enzyme, that uses membrane proton gradient to drive the production of ATP. Talking about actual numbers is difficult here as there is only data available from Rat hepatocytes. Who is to say mammals are representative of all organisms? The estimates of energy of hydrolysis range from ΔG˚ = -48 kJ mol-1 to -30.5 kmol-1. Note that these are considerable, but not exceptional values, so it's easy for many different proteins, that need not be very specialised, to break the bond all over the body. I couldn't even find the numbers for the synthase reaction per ATP, but a single ATP synthase can produce up to 600 ATP per minute.
The final point of this efficiency is that the elements in ATP are very abundant in the biosphere making it easily available. This is ultimately what makes the phosphates a great biomolecule.
ATP is ubiquitous in the body, but in some cases more energy is needed than there are ATP available. In these times of need, ATP can be used to produce more energy, breaking another phosphoanhydride bond to become AMP+2Pi. AMP however is typically a signalling molecule.
With the low activation energy required to break the phosphoanhydride bond, a multitude of enzymes, far too many to list here, can make use of ATP in order to gain energy towards the activation energy for many other functions.