Iterate over range, append string to each

You can try something like this:

//
// helper `forEachRangeCell` function
//

function forEachRangeCell(range, f) {
  const numRows = range.getNumRows();
  const numCols = range.getNumColumns();

  for (let i = 1; i <= numCols; i++) {
    for (let j = 1; j <= numRows; j++) {
      const cell = range.getCell(j, i)

      f(cell)
    }
  }
}

//
// Usage
//

const range = SpreadsheetApp.getActiveSheet().getActiveRange();

forEachRangeCell(range, (cell) => {
  cell.setValue(`${cell.getValue()} string`)
})