What is the smallest valid jpeg file size (in bytes)

A 1x1 grey pixel in 125 bytes using arithmetic coding, still in the JPEG standard even if most decoders can't decode it:

ff d8 : SOI
ff e0 ; APP0
 00 10
 4a 46 49 46 00 01 01 01 00 48 00 48 00 00
ff db ; DQT
 00 43
 00
 03 02 02 02 02 02 03 02
 02 02 03 03 03 03 04 06
 04 04 04 04 04 08 06 06
 05 06 09 08 0a 0a 09 08
 09 09 0a 0c 0f 0c 0a 0b
 0e 0b 09 09 0d 11 0d 0e
 0f 10 10 11 10 0a 0c 12
 13 12 10 13 0f 10 10 10
ff c9 ; SOF
 00 0b
 08 00 01 00 01 01 01 11 00
ff cc ; DAC
 00 06 00 10 10 05
ff da ; SOS
 00 08
 01 01 00 00 3f 00 d2 cf 20
ff d9 ; EOI

I don't think the mentioned 134 byte example is standard, as it is missing an EOI. All decoders will handle this but the standard says it should end with one.

That file can be generated with:

#!/usr/bin/env bash
printf '\xff\xd8' # SOI
printf '\xff\xe0' # APP0
printf  '\x00\x10'
printf  '\x4a\x46\x49\x46\x00\x01\x01\x01\x00\x48\x00\x48\x00\x00'
printf '\xff\xdb' # DQT
printf  '\x00\x43'
printf  '\x00'
printf  '\x03\x02\x02\x02\x02\x02\x03\x02'
printf  '\x02\x02\x03\x03\x03\x03\x04\x06'
printf  '\x04\x04\x04\x04\x04\x08\x06\x06'
printf  '\x05\x06\x09\x08\x0a\x0a\x09\x08'
printf  '\x09\x09\x0a\x0c\x0f\x0c\x0a\x0b'
printf  '\x0e\x0b\x09\x09\x0d\x11\x0d\x0e'
printf  '\x0f\x10\x10\x11\x10\x0a\x0c\x12'
printf  '\x13\x12\x10\x13\x0f\x10\x10\x10'
printf '\xff\xc9' # SOF
printf  '\x00\x0b'
printf  '\x08\x00\x01\x00\x01\x01\x01\x11\x00'
printf '\xff\xcc' # DAC
printf  '\x00\x06\x00\x10\x10\x05'
printf '\xff\xda' # SOS
printf  '\x00\x08'
printf  '\x01\x01\x00\x00\x3f\x00\xd2\xcf\x20'
printf '\xff\xd9' # EOI

and opened fine with GNOME Image Viewer 3.38.0 and GIMP 2.10.18 on Ubuntu 20.10.

Here's an upload on Imgur. Note that Imgur process the file making it larger however if you download it to check, and as seen below, the width=100 image shows white on Chromium 87:

Here's the C++ routine I wrote to do this:

bool is_jpeg(const unsigned char* img_data, size_t size)
{           
    return img_data &&
           (size >= 10) &&
           (img_data[0] == 0xFF) &&
           (img_data[1] == 0xD8) &&
           ((memcmp(img_data + 6, "JFIF", 4) == 0) ||
            (memcmp(img_data + 6, "Exif", 4) == 0));
}

img_data points to a buffer containing the JPEG data.

I'm sure you need more bytes to have a JPEG that will decode to a useful image, but it's a fair bet that if the first 10 bytes pass this test, the buffer probably contains a JPEG.

EDIT: You can, of course, replace the 10 above with a higher value once you decide on one. 134, as suggested in another answer, for example.