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An acid is a substance that donates a proton (H+) to a base. In water, the stronger an acid is, the more it ionizes (splits into ions), and the more H3O+ ions it forms. A base is a substance that accepts a proton from an acid. Similarly, the stronger a base is, the more it ionizes in water and the more OH- ions it forms. For both acids and bases, their strength can be quantified by their respective ionization constants: Ka (acid ionization constant) and Kb (base ionization constant).

In an aqueous solution, water acts as both an acid and a base, as it can donate and accept protons. When water molecules collide, one of them can act as an acid and transfer a proton to the other, which acts as a base. This process creates two ions: H3O+ (hydronium, from the molecule that accepted a proton) and OH- (hydroxide, from the molecule that donated a proton). This phenomenon is called "water ionization" or "autoprotolysis" and can be represented by the reaction: \[2 \mathrm{H}_{2} \mathrm{O} \leftrightarrows \mathrm{H}_{3} \mathrm{O}^{+} + \mathrm{OH}^{-}\] The equilibrium constant for this reaction, called the ion product of water (Kw), is the product of the concentrations of the two ions: \[K_{w} = [\mathrm{H}_{3} \mathrm{O}^{+}][\mathrm{OH}^{-}]\] The value of Kw depends on temperature and is typically around 1.0 x 10^(-14) at 25°C.

In water, the hydronium ion (H3O+) is formed by the ionization of water molecules, as mentioned earlier. Since H3O+ is formed by water accepting a proton, it is considered the strongest acid that can exist in significant amounts in aqueous solutions. Any stronger acid would simply donate its protons to water molecules to form more H3O+ ions, so the presence of an even stronger acid would ultimately lead back to the same hydronium ion. If an acid attempts to transfer a proton to a hydronium ion, the ionized form of the acid would actually have a higher energy state than its neutral form. Thus, strong acids will preferentially donate their protons to water, forming the hydronium ion as the strongest possible acid in these solutions.

In a similar manner as for acids, the hydroxide ion (OH-) is the strongest base that can exist in significant amounts in aqueous solutions. Hydroxide ions are formed by water donating protons. Any stronger base would simply accept the protons from the hydronium ions, ultimately turning back into hydroxide ions. If a base attempts to accept a proton from an OH- ion, the ionized form of the base would have a higher energy state than its neutral form. Therefore, strong bases will preferentially accept protons from water, forming the hydroxide ion as the strongest possible base in these solutions.

In conclusion, the hydronium ion (H3O+) is the strongest acid, and the hydroxide ion (OH-) is the strongest base that can exist in significant amounts in aqueous solutions. This is due to the ionization of water and the equilibrium constants involved. The presence of even stronger acids and bases would lead back to the formation of these ions, as they represent the lowest possible energy states for the protonated and deprotonated forms of water.