...but at least 269 of them appear not to be signifiant enough to be named, only numbered:
Cave
Anstey's Cove Slip Rift No 1
Anstey's Cove Slip Rift No 2
Anstey's Cove Slip Rift No 4
Anstey's Cove Slip Rift No 5
Anstey's Cove Slip Rift No 6
Anstey's Cove Slip Rift No 7
Beer Sea Cave No 1
Beer Sea Cave No 10
Beer Sea Cave No 11
Beer Sea Cave No 12
Beer Sea Cave No 13
Beer Sea Cave No 14
Beer Sea Cave No 15
Beer Sea Cave No 16
Beer Sea Cave No 17
Beer Sea Cave No 18
Beer Sea Cave No 19
Beer Sea Cave No 2
Beer Sea Cave No 20
Beer Sea Cave No 21
Beer Sea Cave No 22
Beer Sea Cave No 23
Beer Sea Cave No 24
Beer Sea Cave No 25
Beer Sea Cave No 26
Beer Sea Cave No 27
Beer Sea Cave No 28
Beer Sea Cave No 29
Beer Sea Cave No 3
Beer Sea Cave No 30
Beer Sea Cave No 31
Beer Sea Cave No 32
Beer Sea Cave No 33
Beer Sea Cave No 34
Beer Sea Cave No 35
Beer Sea Cave No 36
Beer Sea Cave No 37
Beer Sea Cave No 38
Beer Sea Cave No 39
Beer Sea Cave No 4
Beer Sea Cave No 40
Beer Sea Cave No 41
Beer Sea Cave No 42
Beer Sea Cave No 43
Beer Sea Cave No 44
Beer Sea Cave No 45
Beer Sea Cave No 46
Beer Sea Cave No 47
Beer Sea Cave No 5
Beer Sea Cave No 6
Beer Sea Cave No 7
Beer Sea Cave No 8
Beer Sea Cave No 9
Berry Head Quarry Caves. 1
Berry Head Quarry Caves. 2
Berry Head Quarry Caves. 3
Berry Head Quarry Caves. 4
Berry Head Quarry Caves. 5
Berry Head Sea Cave. 1
Berry Head Sea Cave. 2
Berry Head Sea Cave. 3
Berry Head Sea Cave. 4
Bishop's Walk Cave No 1
Bishop's Walk Cave No 2
Chudleigh Cave No 20
Chudleigh Cave No 21
Colesdown Hill Cave No 1
Colesdown Hill Cave No 2
Colesdown Hill Cave No 3
Colesdown Hill Cave No 4
Colesdown Hill Cave No 5
Colesdown Hill Cave No 6
Corbyn's Head Cave No 1
Corbyn's Head Cave No 2
Corbyn's Head Cave No 3
Corbyn's Head Cave No 4
Corbyn's Head Cave No 5
Corbyn's Head Cave No 6
Empacombe cave No 1
Empacombe cave No 2
Empacombe cave No 3
Hexton Quarry Cave 1
Hexton Quarry Cave 2
Hexton Quarry Cave 3
Hoe Sea Cave 1
Hoe Sea Cave 2
Hoe Sea Cave 3
Hoe Sea Cave 4
Hoe Sea Cave 5
Hooe lake Quarry Cave No 1
Hooe lake Quarry Cave No 10
Hooe lake Quarry Cave No 11
Hooe lake Quarry Cave No 12/3/4
Hooe lake Quarry Cave No 15
Hooe lake Quarry Cave No 16
Hooe lake Quarry Cave No 17
Hooe lake Quarry Cave No 18
Hooe lake Quarry Cave No 19/20
Hooe lake Quarry Cave No 2
Hooe lake Quarry Cave No 21
Hooe lake Quarry Cave No 22
Hooe lake Quarry Cave No 23
Hooe lake Quarry Cave No 24
Hooe lake Quarry Cave No 25
Hooe lake Quarry Cave No 26
Hooe lake Quarry Cave No 3
Hooe lake Quarry Cave No 4
Hooe lake Quarry Cave No 5
Hooe lake Quarry Cave No 6
Hooe lake Quarry Cave No 7
Hooe lake Quarry Cave No 8
Hooe lake Quarry Cave No 9
Hopes Nose Submarine cave No 1
Hopes Nose Submarine cave No 2
Hopes Nose Submarine cave No 3
Hopes Nose Submarine cave No 4
Hopes Nose Submarine cave No 5
Hopes Nose Submarine cave No 6
Hopes Nose Submarine cave No 7
Kitley Cave 19
Kitley Cave 22
Kitley Cave 23
Kitley Cave 26
Langford Quarry Cave No 1
Langford Quarry Cave No 10
Langford Quarry Cave No 11
Langford Quarry Cave No 12
Langford Quarry Cave No 2
Langford Quarry Cave No 3
Langford Quarry Cave No 4
Langford Quarry Cave No 5/6
Langford Quarry Cave No 7
Langford Quarry Cave No 8
Langford Quarry Cave No 9
Leek head bay sea cave 1
Leek head bay sea cave 2
Leek head bay sea cave 3
Leek head bay sea cave 4
Leek head bay sea cave 5
Linhay Hill cave No 1
Linhay Hill cave No 2
Linhay Hill cave No 3
Livermead Head Sea Cave No 1
Livermead Head Sea Cave No 2
Livermead Head Sea Cave No 3
Livermead Head Sea Cave No 4
Livermead Head Sea Cave No 5
Livermead Head Sea Cave No 6
Livermead Head Sea Cave No 7
London Bridge Sea Cave (TN) 1
London Bridge Sea Cave (TN) 2
London Bridge Sea Cave (TN) 3
Long Point Quarry Cave No 1
Long Point Quarry Cave No 2
Long Point Quarry Cave No 3
Long Point Quarry Cave No 4
Lundy Island Cave No 1
Lundy Island Cave No 10
Lundy Island Cave No 11
Lundy Island Cave No 12
Lundy Island Cave No 13
Lundy Island Cave No 14
Lundy Island Cave No 15
Lundy Island Cave No 16
Lundy Island Cave No 17
Lundy Island Cave No 18
Lundy Island Cave No 19
Lundy Island Cave No 2
Lundy Island Cave No 20
Lundy Island Cave No 21
Lundy Island Cave No 22
Lundy Island Cave No 23
Lundy Island Cave No 24
Lundy Island Cave No 25
Lundy Island Cave No 26
Lundy Island Cave No 3
Lundy Island Cave No 4
Lundy Island Cave No 5
Lundy Island Cave No 6
Lundy Island Cave No 7
Lundy Island Cave No 8
Lundy Island Cave No 9
Mackerel Cove Sea Cave No 1
Mackerel Cove Sea Cave No 10
Mackerel Cove Sea Cave No 11
Mackerel Cove Sea Cave No 12
Mackerel Cove Sea Cave No 13
Mackerel Cove Sea Cave No 14
Mackerel Cove Sea Cave No 2
Mackerel Cove Sea Cave No 3
Mackerel Cove Sea Cave No 4
Mackerel Cove Sea Cave No 5
Mackerel Cove Sea Cave No 6
Mackerel Cove Sea Cave No 7
Mackerel Cove Sea Cave No 8
Mackerel Cove Sea Cave No 9
Man Sands Sea Cave No 1
Man Sands Sea Cave No 2
Man Sands Sea Cave No 3
Ram's Cliff Cave No 1
Ram's Cliff Cave No 2
Ram's Cliff Cave No 3
Shag Cliff Slip Rift No 1
Shag Cliff Slip Rift No 2
Shag Cliff Slip Rift No 3
Shag Cliff Slip Rift No 4
Shaky Cave
Shaky Cave
Shaky Cave Higher
Shaw's Cave No 10
Shaw's Cave No 11
Shaw's Cave No 12
Shaw's Cave No 13
Shaw's Cave No 14
Shaw's Cave No 15
Shaw's Cave No 16
Shaw's Cave No 17
Shaw's Cave No 2
Shaw's Cave No 20
Shaw's Cave No 3
Shaw's Cave No 4
Shaw's Cave No 5
Shaw's Cave No 6
Smuggler's Hole Cave No 1
Smuggler's Hole Cave No 2
Smuggler's Hole Cave No 3
Totnes Cave (s) No 1
Totnes Cave (s) No 2
Totnes Cave (s) No 3
Watcombe Sea Cave No 1
Watcombe Sea Cave No 2
Watcombe Sea Cave No 3
Watcombe Slip Rift Cave No 1
Watcombe Slip Rift Cave No 2
Watcombe Slip Rift Cave No 3
Watcombe Slip Rift Cave No 4
Watcombe Slip Rift Cave No 5
West Hoe Quarry Cave No 2
West Hoe Quarry Cave No 3
West Hoe Quarry Cave No 4
West Hoe Quarry Cave No 6
West Hoe Quarry Cave No 7
Western Torrs Quarry Cave No 1
Western Torrs Quarry Cave No 10/1
Western Torrs Quarry Cave No 12
Western Torrs Quarry Cave No 13/4
Western Torrs Quarry Cave No 15
Western Torrs Quarry Cave No 16
Western Torrs Quarry Cave No 17
Western Torrs Quarry Cave No 18
Western Torrs Quarry Cave No 19
Western Torrs Quarry Cave No 2/22
Western Torrs Quarry Cave No 20
Western Torrs Quarry Cave No 21
Western Torrs Quarry Cave No 23
Western Torrs Quarry Cave No 24
Western Torrs Quarry Cave No 25
Western Torrs Quarry Cave No 26
Western Torrs Quarry Cave No 27
Western Torrs Quarry Cave No 3
Western Torrs Quarry Cave No 4/17
Western Torrs Quarry Cave No 5
Western Torrs Quarry Cave No 6
Western Torrs Quarry Cave No 7
Western Torrs Quarry Cave No 8
Western Torrs Quarry Cave No 9
Wild Pear Cave No 1
Wild Pear Cave No 2
Wild Pear Cave No 3
I reckon Mendip would have over 1000 caves, if you care to go around numering every rabbit hole