How many chemical reactions occur while baking a cake?

The number of reactions is not as important as understanding the few reactions which are important.

To start with, a leavening agent is needed. Unlike breads which usually use yeast, cakes tend to use chemical leavening agents like baking powder or baking soda (the latter also needs an acid with which to react). There are several kinds of leavening agent, but each produces Carbon dioxide gas. This causes the cake to rise.

Flour contains several proteins, two of which are glutenin and gliadin. These tend to absorb water and then react to form gluten. Stirring the gluten makes it larger and more elastic. This is necessary to trap the Carbon dioxide formed by the leavening.

Oil or butter or margarine is added to help texture and taste. It also seems to play a role in the formation of the gluten.

Eggs are usually added to add solidity and texture to the final cake. The Albumin in the egg whites denatures upon heating. The egg yoke contains additional proteins and also make the mixing of oil or butter with the milk or water easier.

Sugar is added for sweetness. Some recipes call for temperatures which will caramelize at least some of the sugar. The caramelization of table sugar is a very complex reaction.
It depends on what the cake is made of. But assuming you've added eggs, milk, flour etc - there are millions. The denaturing of proteins accounts for most of them as they unfold from their native states at high temperatures. Then there are the simple ones such as the sodium bicarbonate - or baking soda that reacts with the water in the ingredients to make it rise, the proteins in the egg yolk and the white. The whole emulsions going on there - it's pretty complex. In the end... delicious.

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