What is the greatest product of four adjacent numbers in any direction (up, down, left, right, or diagonally) in the 2020 grid?

This solution scans four rows at a time, saves the max adjacent product and scans the next four rows.

(defparameter *grid*
  '((08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08)
    (49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00)
    (81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65)
    (52 70 95 23 04 60 11 42 69 24 68 56 01 32 56 71 37 02 36 91)
    (22 31 16 71 51 67 63 89 41 92 36 54 22 40 40 28 66 33 13 80)
    (24 47 32 60 99 03 45 02 44 75 33 53 78 36 84 20 35 17 12 50)
    (32 98 81 28 64 23 67 10 26 38 40 67 59 54 70 66 18 38 64 70)
    (67 26 20 68 02 62 12 20 95 63 94 39 63 08 40 91 66 49 94 21)
    (24 55 58 05 66 73 99 26 97 17 78 78 96 83 14 88 34 89 63 72)
    (21 36 23 09 75 00 76 44 20 45 35 14 00 61 33 97 34 31 33 95)
    (78 17 53 28 22 75 31 67 15 94 03 80 04 62 16 14 09 53 56 92)
    (16 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 57)
    (86 56 00 48 35 71 89 07 05 44 44 37 44 60 21 58 51 54 17 58)
    (19 80 81 68 05 94 47 69 28 73 92 13 86 52 17 77 04 89 55 40)
    (04 52 08 83 97 35 99 16 07 97 57 32 16 26 26 79 33 27 98 66)
    (88 36 68 87 57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 69)
    (04 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 36)
    (20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16)
    (20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54)
    (01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48)))

(defun max-r (grid)
  "Returns the maximum product of each rows elements"
  (apply #'max (mapcar #'(lambda (lst) (apply #'* lst))
                       grid)))

(defun max-c (grid)
  "Returns the maximum product of each columns elements"
  ;; Transpose the grid then call max-r
  (max-r (apply #'mapcar #'list grid)))

(defun diag-prod (grid &optional (accum 1))
  "Returns the product of the grids diagonal elements,
starting from the top left to the bottom right.
The function works recursively.  If there is a top-left
element in the grid, it multiplies accum by the element.
Then it calls itself with the top row and first column
of the grid removed"
  (if grid
      (diag-prod (mapcar #'cdr (cdr grid)) (* accum (caar grid)))
      accum))

(defun max-d (grid)
  "Returns the maximum product of each diagonals elements"
  (max (diag-prod grid)
           (diag-prod (mapcar #'reverse grid))))

(defun max-adj (grid)
  "Returns the maximum product of the grids adjacent elements.
Elements are multiplied in rows, columns, and diagonals"
  (max (max-r grid) (max-c grid) (max-d grid)))

(defun max-in-4byX (grid &optional (result 0))
  "Returns the maximum product of 4 adjacent elements in
a grid with 4 rows and a variable number of columns"
  (if (< (length (car grid)) 4)
      result
      (let* ((subgrid (mapcar #'(lambda (lst)
                                                  (subseq lst 0 4))
                                            grid))
                (max-in-subgrid (max-adj subgrid))
                (cur-result (cond ((< result max-in-subgrid) max-in-subgrid)
                                           (t result))))
          (max-in-4byX (mapcar #'cdr grid) cur-result))))

(defun euler-11 (grid &optional (result 0))
  "Calculates the max prod of 4 adjacent elements in the top 4 rows
of a grid, removes the top row and repeats.  Returns the maximum
product of 4 adjacent numbers in the entire grid"
  (if (< (length grid) 4)
      result
      (let* ((top-4-rows (subseq grid 0 4))
               (max-in-top-4 (max-in-4byX top-4-rows))
               (cur-result (cond ((< result max-in-top-4) max-in-top-4)
                                           (t result))))
        (euler-11 (cdr grid) cur-result))))