Expressing numbers using 0, 1, 2, 3, and 4

Question

The least number that cannot be written using the numbers 0, 1, 2, and 3, each exactly once, and any combination of standard arithmetic operations (including factorials) is 41. What is the least such number if the numbers 0, 1, 2, 3, and 4 are allowed?

Allowed operations are addition, subtraction, multiplication, division, factorials, exponents, square roots, decimal points, intermediate non-integer results, concatenation, recurring decimals, and any amount of parentheses and brackets. No other digits besides one of each of 0, 1, 2, 3, and 4.

Answer 1

For a start, here is a list of composing all the numbers up to

100

(I avoided using concatenation and decimal points, as based on the conversation in the comments there is some ambiguity if (or when) they are allowed or not.)

$0=0\times(1+2+3+4)$
$1=1^{0+2+3+4}$
$2=0\times(1+3+4)+2$
$3=0\times(1+2+4)+3$
$4=0\times(1+2+3)+4$
$5=0\times(2+3)+4+1$
$6=0\times(1+3)+4+2$
$7=0\times(1+2)+4+3$
$8=0\times2+4+3+1$
$9=0\times1+4+3+2$
$10=0+1+2+3+4$
$11=0\times2+4+3!+1$
$12=0\times1+4+3!+2$
$13=0+2+4+3!+1$
$14=2\times(0!+1)+4+3!$
$15=0+1+2+3\times4$
$16=0\times2+4\times(3+1)$
$17=4^2+0\times3+1$
$18=4^2+0+3-1$
$19=4^2+0+3\times1$
$20=4^2+0+3+1$
$21=4\times(3+2)+0+1$
$22=4\times(3+2)+0+1$
$23=4\times3\times2+0-1$
$24=4\times3\times2+0\times1$
$25=4\times3\times2+0+1$
$26=4!+3+0\times2-1$
$27=4!+3+0\times2\times1$
$28=4!+3+0\times2+1$
$29=4!+3+0\times1+2$
$30=4!+3+0+1+2$
$31=4!+3!+2\times0+1$
$32=4!+3!+1\times0+2$
$33=4!+3!+0+1+2$
$34=4!+3!+0!+1+2$
$35=3!\times(4+2)+0-1$
$36=3!\times(4+2)+0\times1$
$37=3!\times(4+2)+0+1$
$38=3!\times(4+2)+0!+1$
$39=(3!-0!)\times4\times2-1$
$40=(3!-0!)\times4\times2\times1$
$41=(3!-0!)\times4\times2+1$
$42=(3!+0!)\times(4+2)\times1$
$43=(3!+0!)\times(4+2)+1$
$44=4\times(3!+(2+1)!-0!)$
$45=3\times(4^2-1)+0$
$46=3\times(4^2-1)+0!$
$47=3\times4^2-1+0$
$48=3\times4^2+1\times0$
$49=3\times4^2+1+0$
$50=3\times4^2+1+0!$
$51=3\times(4^2+1)+0$
$52=3\times(4^2+1)+0!$
$53=2\times(4!+3)+0-1$
$54=2\times(4!+3)+0\times1$
$55=2\times(4!+3)+0+1$
$56=2\times(4!+3)+0!+1$
$57=2\times(4!+3+0!)+1$
$58=2\times(4!+3+0!+1)$
$59=2\times(4!+3!)+0-1$
$60=2\times(4!+3!)+0\times1$
$61=2\times(4!+3!)+0+1$
$62=2\times(4!+3!)+0!+1$
$63=2\times(4!+3!+0!)+1$
$64=2\times(4!+3!+0!+1)$
$65=4^3+1^2+0$
$66=4^3+2^1+0$
$67=4^3+2+1+0$
$68=4^3+2+1+0!$
$69=4!\times3-2-1+0$
$70=4!\times3-2+1\times0$
$71=4!\times3-2+1+0$
$72=4!\times3+2\times1\times0$
$73=4!\times3+2\times0+1$
$74=4!\times3+2+1\times0$
$75=4!\times3+2+1+0$
$76=(4!+1)\times3+2-0!$
$77=(4!+1)\times3+2+0$
$78=(4!+1)\times3+2+0!$
$79=(4+1)!\times\frac23-0!$
$80=(4+1)!\times\frac23+0$
$81=(4+1)!\times\frac23+0!$
$82=3^4+2-1+0$
$83=3^4+2+1\times0$
$84=3^4+2+1+0$
$85=3^4+2+1+0!$
$86=3^4+(2+1)!-0!$
$87=3^4+(2+1)!+0$
$88=3^4+(2+1)!+0!$
$89=(4!-2)\times(3+1)+0!$
$90=(3+2)!\times(0!-\frac14)$
$91=(4!+(1+2)!)\times3+0!$
$92=(4!-1)\times(3+2-0!)$
$93=3\times(2^{4+1}-0!)$
$94=4!\times(3+0!)-1\times2$
$95=4!\times(3+0!)-1^2$
$96=4!\times(3+0!)\times1^2$
$97=4!\times(3+0!)+1^2$
$98=(4!+1)\times(3+0!)-2$
$99=3\times(2^{4+1}+0!)$
$100=(3!+4)^{2^{1^0}}$

Answer 2

The integers from 1 to 174, and 176, can be so expressed without the need for decimal point or factorial. 175 and 177 can't, though.

elias's answer shows expressions for some integers. In some cases their expression involved factorial or power; it turns out that many integers can be reached using only addition, subtraction and multiplication:

$11=3*4-2+1+0$; $12=2*4+3+1+0$; $13=3*4+2-1+0$; $14=3*4+1*2+0$; $15=3*4+2+1+0$; $16=4*(3+2-1+0)$; $17=3*(4+2)-1+0$; $18=1*3*(4+2+0)$; $19=3*(4+2)+1+0$; $26=2*(3*4+1+0)$; $27=3*(4*2+1+0)$; $28=4*(2*3+1+0)$; $30=2*3*(4+1+0)$; $32=2*4*(3+1+0)$; $36=3*(2+1+0)*4$;

Reaching some integers without using factorial, power or decimal point entails using concenation:

$29=32-4+1+0$; $31=34-2-1+0$; $33=34-2+1+0$; $34=1*34+0*2$; $35=34+2-1+0$; $37=34+2+1+0$; $38=42-3-1+0$; $39=42-1*3+0$; $40=42-3+1+0$; $41=1*(43-2+0)$; $42=43-2+1+0$; $43=1*43+0*2$; $44=43+2-1+0$; $45=1*43+2+0$; $46=43+2+1+0$; $47=41+3*2+0$; $48=3*(12+4+0)$; $49=42+10-3+0$; $50=4*13-2+0$; $51=4*12+3+0$; $52=4*13+0*2$; $53=3*(20-1)-4$; $54=4*13+2+0$; $55=43+12+0$; $56=(3+4)*(10-2)$; $57=(2+4)*10-3$; $58=2*31-4+0$; $59=3*21-4+0$; $60=4*(12+3+0)$; $61=103-42$; $62=(4-2+0)*31$; $63=3*21+0*4$; $64=43+21+0$; $65=3*20+4+1$; $66=2*31+4+0$; $67=2*34-1+0$; $68=1*2*34+0$; $69=2*34+1+0$; $70=2*(34+1+0)$; $71=3*24-1+0$; $72=1*3*24+0$; $73=3*24+1+0$; $74=2*(10*4-3)$; $75=3*(24+1+0)$; $76=2*(41-3+0)$; $77=2*4*10-3$; $78=(4+2+0)*13$; $79=2*41-3+0$; $80=1*240/3$; $81=4*21-3+0$; $82=2*41+0*3$; $83=2*4*10+3$; $84=(4+3+0)*12$; $85=2*43-1+0$; $86=1*2*43+0$; $87=2*43+1+0$; $88=4*(23-1+0)$; $90=(4+3+2)*10$; $91=4*23-1+0$; $92=1*4*23+0$; $93=4*23+1+0$; $94=(2+1)*30+4$; $95=102-4-3$; $96=4*(23+1+0)$; $97=103-4-2$; $98=104-2*3$; $99=104-3-2$; $100=10*(2*3+4)$; $101 = 102+3-4$; $102 = (2+1+0)*34$; $103 = 102+4-3$; $104 = 2*4*(13+0)$; $105 = 103+4-2$; $106 = 2*(43+10)$; $108 = 120-3*4$; $109 = 102+4+3$; $110 = 10*(2*4+3)$; $111 = 103+2*4$; $112 = 4*(3*10-2)$; $113 = 4*(30-2)+1$; $114 = 102+3*4$; $115 = (4+1+0)*23$; $116 = 4*(31-2+0)$; $117 = 3*(41-2+0)$; $118 = 4*32-10$; $119 = 123-4+0$; $120 = (4-3)*120$; $121 = 124-3+0$; $122 = 4*31-2+0$; $123 = 3*(42-1+0)$; $124 = 4*(32-1+0)$; $125 = 3*42-1+0$; $126 = 1*3*42+0$; $127 = 4*32-1+0$; $128 = (1+0)*4*32$; $129 = 4*32+1+0$; $132 = 4*(32+1+0)$; $133 = (4+3)*(20-1)$; $134 = 134+0*2$; $135 = (4 \frac12)*30$; $136 = 132+4+0$; $138 = 4*32+10$; $139 = 142-3+0$; $140 = 2*(4+3)*10$; $141 = 143-2+0$; $142 = 142+0*3$; $144 = 3*4*(12+0)$; $145 = 142+3+0$; $146 = 140+2*3$; $147 = (3+4)*21+0$; $148 = (4+1)*30-2$; $150 = (4+2-1)*30$; $151 = 304/2-1$; $152 = 1*304/2$; $153 = 304/2+1$; $154 = 134+20$; $155 = 20*31/4$; $156 = 3*(42+10)$; $157 = 314/2+0$; $158 = (3+1)*40-2$; $159 = 310/2+4$; $160 = (4+1+0)*32$; $162 = (3+1)*40+2$; $163 = 143+20$; $167 = 201-34$; $168 = (3+1+0)*42$; $169 = 340/2-1$; $170 = 10*34/2$; $171 = 340/2+1$; $172 = 142+30$; $173 = 204-31$; $174 = (4+2)*(30-1)$; $176 = 210-34$

Reaching some integers without factorial or decimal point entails using a power:

$89 = 3^4+10-2$; $107 = 10^2+4+3$; $130 = 2*(4^3+1+0)$; $131 = 132-4^0$; $137 = 201-4^3$; $149 = 140+3^2$; $161 = 2*3^4-1+0$; $164 = 2*(3^4+1+0)$; $165 = 13^2-4+0$; $166 = 102+4^3$;

175 and 177 can be expressed by dint of using a decimal point, but cannot be obtained using just addition, subtraction, multiplication, division, powers and concatenation.

$175 = (34+1)/.2$; $177 = (2+4-.1)*30$

Even if decimal point is allowed, 178 cannot be obtained.