Four 4's

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Yes, that was my intention. And, as it has turned out, it was the intention in the original problem whenever it was first formulated.

Deep sigh. Mark Twain once said, "Past generations have stolen my best ideas." There's a site which gives neat expansions for 1 ≤ N ≤ 100.

The site's been deleted, it looks like...

So we're up to 50 so far amirite?

I can get 51 with the operations used so far. For 53, 55 and 57, I can get it if I use subfactorial (!4 = 9).

51: 4! + 4! + dr(4!)/sqrt(4)
52: 44 + 4 + 4
53: 44 + sqrt(!4)*sqrt(!4)
54: 4! + 4! + 4 + sqrt(4)
55: 4*4*4 - !4
56: 4! + 4! + 4 + 4
57: 44 + 4 + !4

An alternate way at arriving at 55...

55: (4.4/4%)/√4

58: (√4/4%) +4 +4
59: 44 + !4 +dr(4!)
60: (!4/√4%) + !4 +dr(4!)

61: (√4)^dr(4!) - dr(4!)/√4
62: 4*4*4 - √4
63: (√4)^dr(4!) - 4/4
64: 4*4*(√4)*(√4)
65: (√4)^dr(4!) + 4/4
66: 4*4*4 + √4
67: (√4)^dr(4!) + dr(4!)/√4
68: 4*4*4 + 4
69: (√4)^dr(4!) + !4 - 4
70: 4*4*4 + dr(4!)

71: !4*!4 - 4 - dr(4!)
72: !4*!4 - √(!4) - dr(4!)
73: !4*!4 - 4 - 4
74: !4*!4 - dr(4x4)
75: !4*!4 - 4 - √4
76: !4*!4 - !4 + 4
77: !4*!4 - √4 - √4
78: !4*!4 - !4 + dr(4!)
79: !4*!4 + √4 - 4
80: !4*!4 - (4/4)

What is dr?

ruveyn

81: √(!4)*√(!4)*√(!4)*√(!4)

With these, 82-91 come easy just by switching signs:

82: !4*!4 + 4/4
83: !4*!4 - √4 + 4
84: !4*!4 + !4 - dr(4!)
85: !4*!4 + √4 + √4
86: !4*!4 + !4 - 4
87: !4*!4 + 4 + √4
88: !4*!4 + dr(4*4)
89: !4*!4 + 4 + 4
90: !4*!4 + √(!4) + dr(4!)
91: !4*!4 + 4 + dr(4!)

Also a much more elegant 88 is 44 + 44.

92: 44*√4 + 4
93: !4*!4 + dr(4!) + dr(4!)
94: 4/4% - √(!4) - √(!4)
95: 4/4% - √(!4) - √4
96: 4/4% - √4 - √4
97: 4/4% - dr(4!)/√4
98: 4/4% + 4 - √4
99: 4/4% - 4/4
100: 4/4% - 4 + 4
101: 4/4% + 4/4
102: 4/4% - 4 + √4
103: 4/4% + dr(4!)/√4
104: 4/4% + √4 + √4
105: 4/4% + √(!4) + √4
106: 4/4% + √(!4) + √(!4)

107: 4/4% + 4 + √(!4)
108: 4/4% + 4 + 4
109: 4/4% + √(!4) + dr(4!)
110: 4/4% + 4 + dr(4!)
111: 4/4% + !4 + √4
112: 4/4% + dr(4!) + dr(4!)
113: 4/4% + !4 + 4
114: 4,4/4% + 4
115: 4/4% + !4 + dr(4!)
116: 4/4% + (4 * 4)
117: ((4/√4%) * dr(4!)) - √(!4)
118: ((4/√4%) * dr(4!)) - √4
119: 4,4/4% + !4
120: 4/4% + 4/√4%
121: (4!/√4%) + (4/4)
122: (4^4)/√4 - dr(4!)
123: (4!/√4%) + !4 - dr(4!)
124: (4^4)/√4 - 4
125: ((4^4) - dr(4!))/√4
126: (4^4)/√4 - √4
127: ((4^4) - √4)/√4
128: (4^4)/(4 - √4)
129: ((4^4)+√4)/√4
130: ((4^4)+4)/√4
131: ((4^4)+dr(4!))/√4
132: dr(4!)/4% - !4 - !4
133: (4!/√4%) +!4 +4
134: (4!/√4%) +(√!4*4)
135: (4!/√4%) +!4 + dr(4!)
136: (4!/√4%) +(4*4)
137: (4^4)/√4 + !4
138: (4!/√4%) +!4 +!4
139: (dr(4!)/4%) - !4 - √4
140: 4!*dr(4!) - √4 - √4
141: 4!*dr(4!) - !4 + dr(4!)
142: 4!*dr(4!) - 4 + √4
143: 4!*dr(4!) - (4/4)
144: 4!*dr(4!) + 4 - 4
145: 4!*dr(4!) + (4/4)
146: 4!*dr(4!) + 4 - √4
147: 4!*dr(4!) + !4 - dr(4!)
148: 4!*dr(4!) + √4 + √4
149: 4!*dr(4!) + !4 - 4
150: 4!*dr(4!) + 4 + √4
151: 4!*dr(4!) + !4 - √4
152: 4!*dr(4!) + 4 + 4
153: 4!*dr(4!) + (√!4*√!4)
154: 4!*dr(4!) + dr(4!) + 4
155: 4!*dr(4!) + !4 + √4
156: 4!*dr(4!) + !4 + √!4
157: 4!*dr(4!) + !4 + 4
158: 4!*dr(4!) + !4 + 4
159: dr(4!)/4% + (√!4*√!4)
160: 4 * (44 - 4)