First thing I did was recreate the grid in Penpa (added it to the question in case others want to use it). I solved all 4 grids simultaneously and started with the 1s, and single cell is a 1, and this can give info for surrounding cages. In fact, all the 1s in the right most grid can be completely filled in through simple logic:
The best way to start with Sugurus is to look at cages that surround other cages, as the cells that 'see' all cells in another cage must be greater than the greatest number in the surrounded cell. As a result, any cage which has 1 cell that sees all cells in another cage, has very few options. We can place some 4s and 5s throughout as a result, and fill some cages in on the left and third grid, as well as adding a few more 1s.
- Solving the right
A 2 in one of the bottom cages of 5 lets us fill in the third column in the final grid, and then working from the top down the entire grid can be filled in with some simple logic, and just like that one grid is down!
- Feeling lonely
The middle grids look a bit sparse so lets fill some in there. I'd missed an obvious 1-2 at the top of the 2nd grid, which leads to a couple more numbers being placed. The square top right must have a 4 bottom left, which has a domino effect on the rest of the grid. In the first grid, a few 3s can also be placed, which solves a decent chunk.
- Suguru Sandwich
Lets complete the outside grids and finish grid 1 off. The 2 for the star shaped cage of 5 can only go in 1 cell, which leads to a couple numbers being placed and the bottom left 3 being completed. This 3 is the key for the rest of the grid, and everything falls into place.
- 2 down, 2 to go
Lets finish off the 2nd grid now. An important 3 can be placed adjacent to the 4 and 5, which finishes a cage of 4 and leads to 2s being placed throughout. Some other 1s and 4s can be placed, and eventually the grid can be completed fairly simple.
- Final domino!
Finally, onto grid 3. A 2 and a 4 can be placed at the bottom, which solves most of the bottom. A 2 can also be placed top left, which solves the square, and this allows everything to fall into place to get the final answer!