Created in 1922 as part of the direct route between Andover and Warminster avoiding Salisbury, the A344 was shortened twice in the 20th century until its final route was scarcely more than a mile long. The number has now disappeared completely from the maps.
The A344 used roads that pre-date human habitation of Britain, such as the Harroway and the road used to bring the stones to Stonehenge. However, we'll skip forward to classification in 1922.
On classification the A344 ran from the A342 at Weyhill (near Andover) through to Heytesbury, east of Warminster on the A36. In 1934, the A303 was created, taking over the stretch east of Stonehenge. The A360 had became dominant in its merger with the A344 just to the west of Stonehenge by 1956, but the section west of that wasn't downgraded to B390 until the 1970s.
The 1932 route of the A344, compared with a trace of the (much shorter) road 70 years later.
This was merely a two-mile road cutting off a corner, hardly meriting a three-digit number. It started on the A303 at Stonehenge Bottom and finished at Airmans Corner where it ran into the A360, TOTSOing the latter road. Most of it had a 40mph limit with double yellow lines, which is somewhat out of character for a typical rural road.
The remains of the A344 in 2013
The A344 spent several summers as a dead end, with the road blocked off at both Stonehenge Bottom and the Stonehenge Visitors Centre. On the 24th of June 2013, the road was officially closed between those two points. The rest of the A344 was still referred to as such by Wiltshire Council as late as March 2017, but the road is now listed as C506.