REF/PERF: ArrowStringArray.__setitem__
- [x] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests) if fixing a bug or adding a new feature
- [x] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [x] Added an entry in the latest `doc/source/whatsnew/v1.5.0.rst` file if fixing a bug or adding a new feature.

This PR improves the performance of `ArrowStringArray.__setitem__` and avoids some existing behavior that can lead to "over-chunking" of the underlying pyarrow `ChunkedArray` as well as exponential performance.

```
import pandas as pd
import pandas._testing as tm
import pyarrow as pa

ca = pa.chunked_array([
    tm.rands_array(3, 1_000),
    tm.rands_array(3, 1_000),
    tm.rands_array(3, 1_000),
])

num_chunks = []
for n in [1000, 2000, 3000]:
  arr = pd.arrays.ArrowStringArray(ca)
  %timeit arr[:n] = "foo"
  num_chunks.append(arr._data.num_chunks)

print("num_chunks:", num_chunks)
```

```
# main
1.72 s ± 142 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
7.23 s ± 761 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
17.5 s ± 1.26 s per loop (mean ± std. dev. of 7 runs, 1 loop each)
num_chunks: [1002, 2001, 3001]
```

```
# PR
88.5 µs ± 1.12 µs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)
145 µs ± 2.55 µs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)
199 µs ± 3.54 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)
num_chunks: [3, 3, 3]
```

ASVs added:

```
       before           after         ratio
     <main>           <arrow-setitem>
-      42.6±0.7ms       28.7±0.6ms     0.67  array.ArrowStringArray.time_setitem(True)
-      22.5±0.3ms       9.86±0.2ms     0.44  array.ArrowStringArray.time_setitem(False)
-     6.84±0.06ms          430±6μs     0.06  array.ArrowStringArray.time_setitem_list(False)
-      17.4±0.1ms          403±5μs     0.02  array.ArrowStringArray.time_setitem_list(True)
-      1.10±0.01s      3.51±0.04ms     0.00  array.ArrowStringArray.time_setitem_slice(True)
-      1.05±0.02s          259±3μs     0.00  array.ArrowStringArray.time_setitem_slice(False)
```

Does this do anything to help with #45419? > Does this do anything to help with #45419?

No, it does not. If it is decided to make these immutable, then this isn't needed. thanks @lukemanley very nice!

would take a PR to pushed a lot of these indexing methods to a mixin / int of ArrowExtensionArray as @mroeschke is planning on using chunked arrays for backing a new numeric arrow extension array. > would take a PR to pushed a lot of these indexing methods to a mixin / int of ArrowExtensionArray as @mroeschke is planning on using chunked arrays for backing a new numeric arrow extension array.

@jreback - sounds good. Would you suggest a separate mixin for the generic indexing methods or would you just add those to ArrowExtensionArray? IMO if this indexing code is generally applicable to a pyarrow chunked array with any dtype I think it would be best located on ArrowExtensionArray cc @jbrockmendel @mroeschke for comments on that

prob ok to add directly to ArrowExtensionArray i think