1998:

1:     1^998
2:     1^9+9-8
3:     1*(sqrt(9))*(9-8)
4:     1*(sqrt(9))+(9-8)
5:     1+(sqrt(9))+9-8
6:     1^9-(sqrt(9))+8
7:     -1^99+8
8:     1^99*8
9:     1^99+8
10:     1+9*(9-8)
11:     1+9+9-8
12:     19+[sqrt(sqrt(9))]-8
13:     19-((sqrt(9))!)*[sqrt(sqrt(8))]    or     -1^9+((sqrt(9))!)+8
14:     1^9*((sqrt(9))!)+8
15:     1^9+((sqrt(9))!)+8
16:     [19/9]*8
17:     1^9*(9+8)
18:     1^9+(9+8)
19:     -1+(sqrt(9))+9+8
20:     1*(sqrt(9))+9+8
21:     1+(sqrt(9))+9+8
22:     1-(sqrt(9))+(sqrt(9))*8
23:     -1^9+(sqrt(9))*8
24:     19-(sqrt(9))+8
25:     1^9+(sqrt(9))*8
26:     -1+(sqrt(9))+(sqrt(9))*8
27:     1*(sqrt(9))+(sqrt(9))*8
28:     1+(sqrt(9))+(sqrt(9))*8
29:     1+((sqrt(9))!)*((sqrt(9))!)-8
30:     19+(sqrt(9))+8
31:     1*[sqrt(998)]
32:     1+[sqrt(998)]
33:     1*9+(sqrt(9))*8
34:     (-1+(sqrt(9)))*(9+8)
35:     1*9*(sqrt(9))+8
36:     1+9*(sqrt(9))+8
37:     -1+((sqrt(9))!)*((sqrt(9))!)+[sqrt(8)]
38:     1*((sqrt(9))!)*((sqrt(9))!)+[sqrt(8)]
39:     1+((sqrt(9))!)*((sqrt(9))!)+[sqrt(8)]
40:     1-9+((sqrt(9))!)*8
41:     -1-((sqrt(9))!)+((sqrt(9))!)*8
42:     -1*((sqrt(9))!)+((sqrt(9))!)*8
43:     1-((sqrt(9))!)+((sqrt(9))!)*8
44:     -1-(sqrt(9))+((sqrt(9))!)*8
45:     -1+9*((sqrt(9))!)-8
46:     1*9*((sqrt(9))!)-8
47:     1+9*((sqrt(9))!)-8
48:     1*((sqrt(9))+(sqrt(9)))*8
49:     1+((sqrt(9))+(sqrt(9)))*8
50:     -1+(sqrt(9))*(9+8)
51:     1*(sqrt(9))*(9+8)
52:     1+(sqrt(9))*(9+8)
53:     -19+9*8
54:     1*(sqrt(9))*9*(sqrt(8))
55:     1+(sqrt(9))*9*(sqrt(8))
56:     (1+(sqrt(9))+(sqrt(9)))*8
57:     1*9+((sqrt(9))!)*8
58:     1+9+((sqrt(9))!)*8
59:     19*(sqrt(9))+[sqrt(8)]
60:     (1+9)*(sqrt(9))*[(sqrt(8))]
61:     -1+(9*((sqrt(9))!))+8
62:     (1+9)*((sqrt(9))!)+[(sqrt(8))]
63:     (-1*9)+9*8
64:     (-1^9+9)*8
65:     19*(sqrt(9))+8
66:     -1*((sqrt(9))!)+9*8
67:     19+((sqrt(9))!)*8
68:     (1+9)*((sqrt(9))!)+8
69:     -1*(sqrt(9))+9*8
70:     1-(sqrt(9))+9*8
71:     -1^9*9*8
72:     1^9*9*8
73:     1^9+9*8
74:     -1+(sqrt(9))+9*8
75:     1*(sqrt(9))+9*8
76:     1+(sqrt(9))+9*8
77:     (1+((sqrt(9))!)) * ((sqrt(9))+8)
78:     -1 + [sqrt(99) * 8]     or     1*((sqrt(9))!)+9*8
79:     -19+98
80:     -1+9*(9!/8!)
81:     1*9*(9!/8!)
82:     1+9*(9!/8!)
83:     1*9*9+[sqrt(8)]
84:     1+9*9+[sqrt(8)]
85:     (-1+(sqrt(9))!) * (9+8)
86:     -1-(sqrt(9)) + (((sqrt(9))!)!)/8
87:     1*((sqrt(9))!) + 9^[sqrt(8)]
88:     -1+9*9+8
89:     1*9*9+8
90:     1+9*9+8
91:     19+9*8    or     1*99-8
92:     1+99-8
93:     1-(sqrt(9))!+98
94:     -1-sqrt(9)+98
95:     -1*sqrt(9)+98
96:     1-sqrt(9)+98    or     ((1+sqrt(9)) * sqrt(9) * 8
97:     -1^9 + 98
98:     1^9 * 98
99:     1^9 + 98
100:     -1 + sqrt(9) + 98