7 gear to an 8 gear cluster

Imagine you have a splined cassette hub, common these days on newer wheels. Most newer Shimano standard hubs will take at least 7-10 even 11 sprockets these days, give or take inexpensive plastic spacers to define the precise position of a cassette on a hub. Meaning you could slap a 9-speed cassette on your freehub that formerly carried your 7-speed cassette and have your machine converted from an n x 7 to an n x 9.

None of such. There are several reasons why this is just a nice dream.

1. the 7-s chain is wider than the 9-s chain and will rub against and wedge between the two neighbouring sprockets, although quite often 7/8/9-s chains mostly fit either,
2. but to fit the 9 sprockets in the same space as the previously 7, the sprockets are more tightly packed and thinner, 3 meaning that the motion of the RD is smaller, the steps are shorter, implying that both your shifter lever and your RD must be from the same speed-group to work with a 7-s, 8-s or 9-s system.

Conclusion: The three components belong to a family, shifter, cassette and derailleur. The FD is a component of a lesser level, fitting 7, 8, 9 speeds. You can't mix groups in speeds. Shimano does have a data-sheet on a page that informs on (in)compatibilities. It's not the letter of the law, some things may work but others surely won't.